site/page-data/ru/projects/math/page-data.json

{"componentChunkName":"component---src-pages-projects-math-js","path":"/ru/projects/math/","result":{"data":{"ru_projects":{"edges":[{"node":{"html":"<p>Одной из задач, решаемых группой, является популяризация и развитие метода статистической регуляризации, созданного В.Ф. Турчинным в 70-х годах XX века.</p>\n<p>Типичной некорректной обратной задачей, возникающей в физике, является уравнение Фредгольма I рода: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><munderover><mo>∫</mo><mi>a</mi><mi>b</mi></munderover><mi>d</mi><mi>x</mi><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y) = \\int \\limits_a^b dx K(x,y)\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.7083579999999996em;vertical-align:-1.561125em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.1472329999999995em;\"><span style=\"top:-1.8988750000000003em;margin-left:-0.44445em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">a</span></span></span><span style=\"top:-3.361125em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;\">∫</span></span></span><span style=\"top:-4.921125em;margin-left:0.44445em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">b</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.561125em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">x</span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span></span>\n<p>Фактически, это уравнение описывает следующее: аппаратная функция прибора <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">K(x,y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span> действует на иследуемый спектр или иной входной сигнал <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi></mrow><annotation encoding=\"application/x-tex\">\\varphi</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathdefault\">φ</span></span></span></span>,  в результате чего исследователь наблюдает выходной сигнал <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span>. Целью исследователя является востановить сигнал <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi></mrow><annotation encoding=\"application/x-tex\">\\varphi</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathdefault\">φ</span></span></span></span> по известным <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span> и <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">K(x,y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span>. Казалось бы, восстановление сигнала не является сложной задачей, поскольку уравнение Фредгольма имеет точное решение. Но уравение Фредгольма некорректно - бесконечно малое изменение начальных условий приводит к конечному изменению решения. Таким образом, наличие шумов, присутствущих в любом эксперименте, обесценивает попытки решить это уравение <strong>точно</strong>.</p>\n<h3>Теория</h3>\n<p>Рассмотрим некую алгебраизацию уравнения Фредгольма: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>f</mi><mi>m</mi></msub><mo>=</mo><msub><mi>K</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub><msub><mi>φ</mi><mi>n</mi></msub></mrow><annotation encoding=\"application/x-tex\">f_m = K_{mn}\\varphi_n</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8777699999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">m</span><span class=\"mord mathdefault mtight\">n</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>\n<p>С точки зрения математической статистики мы должны должны оценить <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> по реализации <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span>, зная плотность вероятности для <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> и содержимое матрицы <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi></mrow><annotation encoding=\"application/x-tex\">K</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span></span></span></span>. Действуя в духе теории принятия решений, мы должны выбрать вектор-функцию <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{S}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9663299999999999em;vertical-align:0em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span>, определяющую <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> на основе <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> и называемую <em>стратегией</em>. Для того, чтобы определить, какие стратегии более оптимальные, мы введем <em>квадратичную функцию потерь</em>:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>L</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover><mo>−</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><msup><mo>)</mo><mn>2</mn></msup><mo separator=\"true\">,</mo></mrow><annotation encoding=\"application/x-tex\">L(\\hat{\\varphi},\\vec{S}) = (\\hat{\\varphi}-\\vec{S})^2,</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.21633em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">L</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.21633em;vertical-align:-0.25em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mpunct\">,</span></span></span></span></span>\n<p>где <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> — наилучшее решение. Согласно баейсовскому подходу рассмотрим <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> как <strong>случайную переменную</strong> и переместим нашу неопределенность о <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> в <em>априорную плотность</em> <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>, выражающую <strong>достоверность</strong> различных возможных законов природы и определяемую на основе информации, сущетсвующей до проведения эксперимента. При таком подходе выбор оптимальной стратегии основывается на минимизации <em>апостериорного риска</em>:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>r</mi><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover></msub><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>≡</mo><msub><mi>E</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></msub><msub><mi>E</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></msub><mo>[</mo><mi>L</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><mo>)</mo><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>]</mo></mrow><annotation encoding=\"application/x-tex\">r_{\\vec{S}}(\\vec{\\varphi}) \\equiv E_{\\vec{\\varphi}}E_{\\vec{f}}[L(\\vec{\\varphi},\\vec{S})|\\vec{\\varphi}]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0816309999999998em;vertical-align:-0.3316309999999999em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3683690000000004em;margin-left:-0.02778em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-2.96633em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3316309999999999em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≡</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.4418459999999997em;vertical-align:-0.4755159999999998em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.5450000000000004em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">φ</span></span></span><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2911079999999999em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3605920000000005em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-2.97744em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4755159999999998em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mord mathdefault\">L</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span></span>\n<p>Тогда оптимальная стратегия в случае квадратичной функции потерь хорошо известна: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msubsup><mi>S</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msubsup><mo>=</mo><mi>E</mi><mo>[</mo><msub><mi>φ</mi><mi>n</mi></msub><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>]</mo><mo>=</mo><mo>∫</mo><msub><mi>φ</mi><mi>n</mi></msub><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">S^{opt} _n= E[\\varphi_n|\\vec{f}] = \\int \\varphi_n P(\\vec{\\varphi}|\\vec{f})d\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0905559999999999em;vertical-align:-0.247em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8435559999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.1130000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">o</span><span class=\"mord mathdefault mtight\">p</span><span class=\"mord mathdefault mtight\">t</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">[</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<p><em>Апостерионая плотность</em> <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi}|\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> определяется по\nтеореме Баейса: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><mrow><mo>∫</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi}|\\vec{f})= \\frac{P(\\vec{\\varphi})P(\\vec{f}|\\vec{\\varphi})}{\\int d\\vec{\\varphi}P(\\vec{\\varphi})P(\\vec{f}|\\vec{\\varphi})}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.828em;vertical-align:-1.17356em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.65444em;\"><span style=\"top:-2.1325600000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.17356em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>\n<p>Кроме того, такой подход позволяет определить дисперсию полученного решения: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mrow><mo fence=\"true\">⟨</mo><msubsup><mi>σ</mi><mi>n</mi><mn>2</mn></msubsup><mo fence=\"true\">⟩</mo></mrow><mo>=</mo><mo>∫</mo><mo>(</mo><msub><mi>φ</mi><mi>n</mi></msub><mo>−</mo><msubsup><mi>S</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msubsup><msup><mo>)</mo><mn>2</mn></msup><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\left\\langle \\sigma_n^2 \\right\\rangle = \\int (\\varphi_n - S^{opt}_n)^2 P(\\vec{\\varphi}|\\vec{f})d\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2141179999999998em;vertical-align:-0.35001em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">⟨</span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">⟩</span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8435559999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.1130000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">o</span><span class=\"mord mathdefault mtight\">p</span><span class=\"mord mathdefault mtight\">t</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<p>Мы получили решение, введя априорную плотность <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>. Можем ли мы сказать, что-либо о том мире функций <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>, который задается априорной плотностью? Если ответ на этот вопрос отрицательный, то мы должны будем принять все возможные <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> равновероятными и вернуться к нерегуляризованному решению. Таким образом, мы должны ответить на этот вопрос положительно. Именно в этом заключается метод статистической регуляризации — регуляризация решения за счет введения дополнительной априорной информации о <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>. Если исследователь уже обладает какой-либо априорной информацией (априорной плотностью <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>), он может просто вычислить интеграл и получить ответ. В случае, если такой информации нет, в следующем параграфе описывается, какой минимальной информацией может обладать исследователь и как её использовать для получения регулязованного решения.</p>\n<h3>Априоная информация</h3>\n<p>Как показали британские ученые, во всем остальном мире любят дифференцировать. Причем, если математик будет задаваться вопросами о правомерности этой операции, то физик оптимистично верит, что законы природы описываются “хорошим” функциями, то есть гладкими. Иначе говоря, он назначает более гладким <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> более высокую априорную плотность вероятности. Так давайте попробуем ввести априорную вероятность, основанную на гладкости. Для этого мы вспомним, что введение априорной иформации — это некоторое насилие над миром, принуждающее законы природы выглядеть удобным для нас образом. Это насилие следует свести к минимуму, и, вводя априорную плотность вероятности, необходимо, что бы <em>информация Шеннона</em> относительно <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>, содержащаяся в <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>, была минимальной. Формализуя выше сказанное, выведем вид априорной плотности, основанной на гладкости функции. Для этого мы будем искать условный экстремум информации: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>I</mi><mo>[</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>]</mo><mo>=</mo><mo>∫</mo><mi>ln</mi><mo>⁡</mo><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>→</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow><annotation encoding=\"application/x-tex\">I[P(\\vec{\\varphi})] = \\int \\ln{P(\\vec{\\varphi})} P(\\vec{\\varphi}) d\\vec{\\varphi} \\to min</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07847em;\">I</span><span class=\"mopen\">[</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">ln</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.65952em;vertical-align:0em;\"></span><span class=\"mord mathdefault\">m</span><span class=\"mord mathdefault\">i</span><span class=\"mord mathdefault\">n</span></span></span></span></span>\n<p>При следующих условиях:</p>\n<ol>\n<li>Условие на гладкость <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>. Пусть <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><annotation encoding=\"application/x-tex\">\\Omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord\">Ω</span></span></span></span> — некоторая матрица, характеризующая гладкость функции. Тогда потребуем, чтобы достигалось определённое значение функционала гладкости: </li>\n</ol>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mo>∫</mo><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\int (\\vec{\\varphi},\\Omega\\vec{\\varphi}) P(\\vec{\\varphi}) d\\vec{\\varphi} = \\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span></span>\n<p>Внимательный читатель должен задать вопрос об определении значения параметра <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span>. Ответ на него будет дан далее по тексту.</p>\n<ol start=\"2\">\n<li>Нормированность вероятности на единицу: </li>\n</ol>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mo>∫</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\int P(\\vec{\\varphi}) d\\vec{\\varphi} = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span></span>\n<p>При этих условиях доставлять минимум функционалу будет следующая функция: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>P</mi><mi>α</mi></msub><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mfrac><mrow><msup><mi>α</mi><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup><mi>det</mi><mo>⁡</mo><msup><mi mathvariant=\"normal\">Ω</mi><mrow><mn>1</mn><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup></mrow><mrow><mo>(</mo><mn>2</mn><mi>π</mi><msup><mo>)</mo><mrow><mi>N</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P_{\\alpha}(\\vec{\\varphi})  = \\frac{\\alpha^{Rg(\\Omega)/2}\\det\\Omega^{1/2}}{(2\\pi)^{N/2}} \\exp(-\\frac{1}{2} (\\vec{\\varphi},\\alpha\\Omega\\vec{\\varphi}))</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.519em;vertical-align:-0.954em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.565em;\"><span style=\"top:-2.2960000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">π</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.814em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.10903em;\">N</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8879999999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">det</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\">Ω</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8879999999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.954em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span></span>\n<p>Параметр <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> cвязан с <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span>, но поскольку у нас нет собственно информации о конкректных значениях функционала гладкости, выяснять, как именно он связан, бессмысленно. Что же тогда делать с <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, спросите вы? Здесь перед вами расрываются три пути: </p>\n<ol>\n<li>подбирать значение параметра <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> вручную, и тем самым перейти к регуляризации Тихонова </li>\n<li>усреднить по всем возможным <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, предпологая все возможные <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> равновероятными</li>\n<li>выбрать наиболее вероятное <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> по его апостериорной плотности вероятности <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mi>α</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\alpha|\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>. Этот подход верен, если мы предполагаем, что в экспериментальных данных содержится достаточно информации об <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>. </li>\n</ol>\n<p>Первый случай нам мало интересен. Во втором случае мы получим следующую формулу для решения:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mrow><mo fence=\"true\">⟨</mo><msub><mi>φ</mi><mi>i</mi></msub><mo fence=\"true\">⟩</mo></mrow><mo>=</mo><mfrac><mrow><mo>∫</mo><mi>d</mi><mi>φ</mi><mtext>&ThinSpace;</mtext><msub><mi>φ</mi><mi>i</mi></msub><mi>P</mi><mo>(</mo><mi>f</mi><mi mathvariant=\"normal\">∣</mi><mi>φ</mi><mo>)</mo><mo>∫</mo><mi>d</mi><mi>α</mi><mtext>&ThinSpace;</mtext><mi>P</mi><mo>(</mo><mi>α</mi><mo>)</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow><mrow><mo>∫</mo><mi>d</mi><mi>φ</mi><mi>P</mi><mo>(</mo><mi>f</mi><mi mathvariant=\"normal\">∣</mi><mi>φ</mi><mo>)</mo><mo>∫</mo><mi>d</mi><mi>α</mi><mtext>&ThinSpace;</mtext><mi>P</mi><mo>(</mo><mi>α</mi><mo>)</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">\\left\\langle \\varphi_i \\right\\rangle = \\frac{\\int d\\varphi\\, \\varphi_i P(f|\\varphi) \\int\\limits d\\alpha\\,P(\\alpha)  \\alpha^{\\frac{Rg(\\Omega)}{2}} \\exp(-\\frac{\\alpha}{2} (\\vec{\\varphi},\\Omega\\vec{\\varphi}))}{\\int d\\varphi P(f|\\varphi) \\int\\limits d\\alpha\\,P(\\alpha)  \\alpha^{\\frac{Rg(\\Omega)}{2}} \\exp(-\\frac{\\alpha}{2} (\\vec{\\varphi},\\Omega\\vec{\\varphi}))}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\">⟨</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.13445em;vertical-align:-1.31em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8244500000000001em;\"><span style=\"top:-2.1244500000000004em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">φ</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord\">∣</span><span class=\"mord mathdefault\">φ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.075em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.695392em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span><span style=\"top:-3.3194500000000002em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.8244500000000006em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">φ</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord\">∣</span><span class=\"mord mathdefault\">φ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.08945em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.695392em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.31em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>\n<p>Третий случай будет рассмотрен в следующем разделе на примере гауссовых шумов в эксперименте.</p>\n<h3>Случай гауссовых шумов</h3>\n<p>Случай, когда ошибки в эксперименте распределены по Гауссу, замечателен тем, что можно получить аналитическое решение нашей задачи. Решение и его ошибка будут иметь следующий вид:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><msup><mi>α</mi><mo>∗</mo></msup><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><msup><mn>1</mn><mi>T</mi></msup></mrow></msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}  = (K^T\\Sigma^{-1}K +\\alpha^*\\Omega)^{-1}K^T\\Sigma^{-1^{T}}\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1413309999999999em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.306365em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.738696em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.056365em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9190928571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi mathvariant=\"normal\">Σ</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></msub><mo>=</mo><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><msup><mi>α</mi><mo>∗</mo></msup><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding=\"application/x-tex\">\\Sigma_{\\vec{\\varphi}}  = (K^T\\Sigma^{-1}K+\\alpha^*\\Omega)^{-1}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9744379999999999em;vertical-align:-0.2911079999999999em;\"></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.5450000000000004em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">φ</span></span></span><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2911079999999999em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1413309999999999em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141079999999999em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.738696em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span></span></span></span></span>\n<p>где <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi mathvariant=\"normal\">Σ</mi></mrow><annotation encoding=\"application/x-tex\">\\Sigma</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord\">Σ</span></span></span></span> - ковариационная матрица многомерного распределения Гаусса, <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msup><mi>α</mi><mo>∗</mo></msup></mrow><annotation encoding=\"application/x-tex\">\\alpha^*</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.688696em;vertical-align:0em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.688696em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span></span></span></span> - наиболее вероятное значение параметра <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, которое определяется из условия максимума апостериорной плотности вероятности: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mi>α</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mi>C</mi><mn>39</mn><mo separator=\"true\">;</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><msqrt><mrow><mi mathvariant=\"normal\">∣</mi><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant=\"normal\">∣</mi></mrow></msqrt><mi>exp</mi><mo>⁡</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><msup><mn>1</mn><mi>T</mi></msup></mrow></msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\alpha|\\vec{f}) = C39; \\alpha^{\\frac{Rg(\\Omega)}{2}}\\sqrt{|(K^T\\Sigma^{-1}K+\\alpha\\Omega)^{-1}|}\\exp(\\frac{1}{2} \\vec{f}^T\\Sigma^{-1}K^{T}(K^T\\Sigma^{-1}K+\\alpha\\Omega)^{-1}K^T\\Sigma^{-1^{T}}\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">C</span><span class=\"mord\">3</span><span class=\"mord\">9</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.13945em;\"><span style=\"top:-3.4130000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2925405em;\"><span class=\"svg-align\" style=\"top:-3.8em;\"><span class=\"pstrut\" style=\"height:3.8em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mord\">∣</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.767331em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\">∣</span></span></span><span style=\"top:-3.2525405em;\"><span class=\"pstrut\" style=\"height:3.8em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:1.8800000000000001em;\"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,\n572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,\n-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39\nc-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60\ns208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,\n-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10z\nM1001 80H400000v40H1013z'/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5474595em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.306365em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.056365em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9190928571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>\n<p>В качестве примера рассмотрим востановление спектра, состоящего из двух гауссовых пиков, которые попали под действие интегрального ядра-ступеньки (функции Хевисайда).</p>\n<img src=\"/images/projects/math/deconvolution.png\" alt=\"deconvolution\"/>","frontmatter":{"shortTitle":"Обратные задачи","title":"Статистическая регуляризация некорректных обратных задач","id":"deconvolution"}}},{"node":{"html":"<table>\n    <tbody><tr><td>\n        <div classname=\"col-lg-9\">\n            <img src=\"/images/projects/math/gears_animated.gif\" alt=\"Under construction...\">\n        </div>\n    </td>\n    <td>\n        <div classname=\"col-lg-8\" align=\"center\"><h3>&#x42D;&#x442;&#x43E;&#x442; &#x440;&#x430;&#x437;&#x434;&#x435;&#x43B; &#x434;&#x43E;&#x440;&#x430;&#x431;&#x430;&#x442;&#x44B;&#x432;&#x430;&#x435;&#x442;&#x441;&#x44F;...</h3></div>\n    </td>\n</tr></tbody></table>","frontmatter":{"shortTitle":"Функции значимости","title":"Оптимальное планирование эксперимента при помощи функций значимости параметров","id":"significance"}}}]},"en_projects":{"edges":[{"node":{"html":"<p>One of the tasks solved by the group is the popularization and development of the statistical regularization method created by V.F. Turchin in the 70s of the XX century.</p>\n<p>A typical incorrect inverse problem that arises in physics is the Fredholm equation of the first kind:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><munderover><mo>∫</mo><mi>a</mi><mi>b</mi></munderover><mi>d</mi><mi>x</mi><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y) = \\int \\limits_a^b dx K(x,y)\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.7083579999999996em;vertical-align:-1.561125em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.1472329999999995em;\"><span style=\"top:-1.8988750000000003em;margin-left:-0.44445em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">a</span></span></span><span style=\"top:-3.361125em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;\">∫</span></span></span><span style=\"top:-4.921125em;margin-left:0.44445em;\"><span class=\"pstrut\" style=\"height:3.3600000000000003em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">b</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.561125em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">x</span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span></span>\n<p>In fact, this equation describes the following: the hardware function of the device <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">K(x,y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span> acts on the studied spectrum or other input signal <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi></mrow><annotation encoding=\"application/x-tex\">\\varphi</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathdefault\">φ</span></span></span></span>,  as a result, the researcher observes the output signal <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span>. The aim of the researcher is to restore the signal <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi></mrow><annotation encoding=\"application/x-tex\">\\varphi</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathdefault\">φ</span></span></span></span> from the known <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">f(y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span> and <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi><mo>(</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">K(x,y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span></span></span></span>. It would seem that signal recovery is not a difficult task, since the Fredholm equation has an exact solution. But the Fredholm equation is incorrect - an infinitesimal change in the initial conditions leads to a final change in the solution. Thus, the presence of noise present in any experiment invalidates attempts to solve this equation <strong>for sure</strong>.</p>\n<h3>Theory</h3>\n<p>Consider a certain algebraization of the Fredholm equation:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>f</mi><mi>m</mi></msub><mo>=</mo><msub><mi>K</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub><msub><mi>φ</mi><mi>n</mi></msub></mrow><annotation encoding=\"application/x-tex\">f_m = K_{mn}\\varphi_n</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8777699999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">m</span><span class=\"mord mathdefault mtight\">n</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>\n<p>In terms of mathematical statistics, we must evaluate<span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> using implementation <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span>, knowing the probability density for <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> and matrix <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>K</mi></mrow><annotation encoding=\"application/x-tex\">K</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span></span></span></span> content. Acting in the spirit of decision theory, we must choose a vector function <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{S}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9663299999999999em;vertical-align:0em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span>, defining <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> on base of <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.1718799999999998em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> and called <em>strategy</em>. In order to determine which strategies are more optimal, we introduce the <em>squared loss function</em>:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>L</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover><mo>−</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><msup><mo>)</mo><mn>2</mn></msup><mo separator=\"true\">,</mo></mrow><annotation encoding=\"application/x-tex\">L(\\hat{\\varphi},\\vec{S}) = (\\hat{\\varphi}-\\vec{S})^2,</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.21633em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">L</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.21633em;vertical-align:-0.25em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mpunct\">,</span></span></span></span></span>\n<p>where <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\hat{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\">^</span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> — the best decision. According to the Bayesian approach, we consider <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> as <strong>random variable</strong> and move our uncertainty about <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span> in <em>prior density</em> <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>, Expressing <strong>reliability</strong> of the various possible laws of nature and determined on the basis of information prior to the experiment. With this approach, the choice of an optimal strategy is based on minimizing <em>aposterior risk</em>:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>r</mi><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover></msub><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>≡</mo><msub><mi>E</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></msub><msub><mi>E</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></msub><mo>[</mo><mi>L</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mover accent=\"true\"><mi>S</mi><mo>⃗</mo></mover><mo>)</mo><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>]</mo></mrow><annotation encoding=\"application/x-tex\">r_{\\vec{S}}(\\vec{\\varphi}) \\equiv E_{\\vec{\\varphi}}E_{\\vec{f}}[L(\\vec{\\varphi},\\vec{S})|\\vec{\\varphi}]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0816309999999998em;vertical-align:-0.3316309999999999em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">r</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3683690000000004em;margin-left:-0.02778em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-2.96633em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3316309999999999em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≡</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.4418459999999997em;vertical-align:-0.4755159999999998em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.5450000000000004em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">φ</span></span></span><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2911079999999999em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3605920000000005em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-2.97744em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4755159999999998em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mord mathdefault\">L</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9663299999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span></span></span><span style=\"top:-3.25233em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span></span>\n<p>Then the optimal strategy in case of the square loss function is well known: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msubsup><mi>S</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msubsup><mo>=</mo><mi>E</mi><mo>[</mo><msub><mi>φ</mi><mi>n</mi></msub><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>]</mo><mo>=</mo><mo>∫</mo><msub><mi>φ</mi><mi>n</mi></msub><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">S^{opt} _n= E[\\varphi_n|\\vec{f}] = \\int \\varphi_n P(\\vec{\\varphi}|\\vec{f})d\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0905559999999999em;vertical-align:-0.247em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8435559999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.1130000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">o</span><span class=\"mord mathdefault mtight\">p</span><span class=\"mord mathdefault mtight\">t</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">[</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<p><em>Aposterior density</em> <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi}|\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> is determined by the Bayes theorem: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><mrow><mo>∫</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi}|\\vec{f})= \\frac{P(\\vec{\\varphi})P(\\vec{f}|\\vec{\\varphi})}{\\int d\\vec{\\varphi}P(\\vec{\\varphi})P(\\vec{f}|\\vec{\\varphi})}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.828em;vertical-align:-1.17356em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.65444em;\"><span style=\"top:-2.1325600000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.17356em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>\n<p>In addition, this approach allows us to determine the dispersion of the resulting solution: </p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mrow><mo fence=\"true\">⟨</mo><msubsup><mi>σ</mi><mi>n</mi><mn>2</mn></msubsup><mo fence=\"true\">⟩</mo></mrow><mo>=</mo><mo>∫</mo><mo>(</mo><msub><mi>φ</mi><mi>n</mi></msub><mo>−</mo><msubsup><mi>S</mi><mi>n</mi><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msubsup><msup><mo>)</mo><mn>2</mn></msup><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\left\\langle \\sigma_n^2 \\right\\rangle = \\int (\\varphi_n - S^{opt}_n)^2 P(\\vec{\\varphi}|\\vec{f})d\\vec{\\varphi}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2141179999999998em;vertical-align:-0.35001em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">⟨</span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size1\">⟩</span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8435559999999999em;\"><span style=\"top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n</span></span></span><span style=\"top:-3.1130000000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">o</span><span class=\"mord mathdefault mtight\">p</span><span class=\"mord mathdefault mtight\">t</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.247em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8641079999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<p>We got the solution by introducing a priori density <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>. Can we say anything about the world of <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> functions, which is defined by a priori density? If the answer to this question is no, we will have to accept all possible <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> equally probable and return to the irregular solution. Thus, we should answer this question positively. This is the statistical regularization method - regularization of the solution by introducing additional a priori information about <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>. If a researcher already has some a priori information (a priori density of <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>), he can simply calculate the integral and get an answer. If there is no such information, the following paragraph describes what minimal information a researcher can have and how to use it to obtain a regularized solution.</p>\n<h3>Prior information</h3>\n<p>As British scientists have shown, the rest of the world likes to differentiate. Moreover, if a mathematician will be asked questions about the validity of this operation, the physicist optimistically believes that the laws of nature are described by \"good\" functions, that is, smooth. In other words, he assigns smoother <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> a higher a priori probability density. So let's try to introduce an a priori probability based on smoothness. To do this, we will remember that the introduction of the a priori probability is some kind of violence against the world, forcing the laws of nature to look comfortable for us. This violence should be minimized, and by introducing an a priori probability density, it is necessary that _ Shannon_'s information regarding <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span> contained in <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\vec{\\varphi})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> be minimal. Formalizing the above, let us derive a type of a priori density based on the smoothness of the function. For this purpose, we will search for a conditional extremum of information:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>I</mi><mo>[</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>]</mo><mo>=</mo><mo>∫</mo><mi>ln</mi><mo>⁡</mo><mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo></mrow><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>→</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow><annotation encoding=\"application/x-tex\">I[P(\\vec{\\varphi})] = \\int \\ln{P(\\vec{\\varphi})} P(\\vec{\\varphi}) d\\vec{\\varphi} \\to min</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07847em;\">I</span><span class=\"mopen\">[</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">ln</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">→</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.65952em;vertical-align:0em;\"></span><span class=\"mord mathdefault\">m</span><span class=\"mord mathdefault\">i</span><span class=\"mord mathdefault\">n</span></span></span></span></span>\n<p>Under the following conditions:</p>\n<ol>\n<li>Condition for smoothness <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">\\varphi(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\">φ</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\">x</span><span class=\"mclose\">)</span></span></span></span>. Let <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi mathvariant=\"normal\">Ω</mi></mrow><annotation encoding=\"application/x-tex\">\\Omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord\">Ω</span></span></span></span> be some matrix characterizing the smoothness of the function. Then we demand that a certain value of the smoothness functional is achieved:</li>\n</ol>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mo>∫</mo><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\int (\\vec{\\varphi},\\Omega\\vec{\\varphi}) P(\\vec{\\varphi}) d\\vec{\\varphi} = \\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span></span>\n<p>The attentive reader should ask a question about the definition of <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span>. The answer to this question will be given further down the text.</p>\n<ol start=\"2\">\n<li>The normality of probability per unit: </li>\n</ol>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mo>∫</mo><mi>P</mi><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mi>d</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\int P(\\vec{\\varphi}) d\\vec{\\varphi} = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.22225em;vertical-align:-0.86225em;\"></span><span class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathdefault\">d</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span></span>\n<p>Under these conditions, the following function will deliver a minimum to the function:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi>P</mi><mi>α</mi></msub><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mfrac><mrow><msup><mi>α</mi><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup><mi>det</mi><mo>⁡</mo><msup><mi mathvariant=\"normal\">Ω</mi><mrow><mn>1</mn><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup></mrow><mrow><mo>(</mo><mn>2</mn><mi>π</mi><msup><mo>)</mo><mrow><mi>N</mi><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow></msup></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P_{\\alpha}(\\vec{\\varphi})  = \\frac{\\alpha^{Rg(\\Omega)/2}\\det\\Omega^{1/2}}{(2\\pi)^{N/2}} \\exp(-\\frac{1}{2} (\\vec{\\varphi},\\alpha\\Omega\\vec{\\varphi}))</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.519em;vertical-align:-0.954em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.565em;\"><span style=\"top:-2.2960000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">π</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.814em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.10903em;\">N</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8879999999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">det</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\">Ω</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8879999999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mord mtight\">/</span><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.954em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span></span>\n<p>The <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> parameter is associated with <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding=\"application/x-tex\">\\omega</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">ω</span></span></span></span>, but since we don't actually have information about the specific values of the smoothness functionality, it makes no sense to find out how it is associated. Then what to do with <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, you ask? There are three paths: </p>\n<ol>\n<li>select the value of the parameter <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> manually, and thus proceed to regularization of Tikhonov </li>\n<li>average all possible <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, assuming all possible <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> equally probable</li>\n<li>choose the most likely <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> by its a posteriori probability density of <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mi>α</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\alpha|\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>. This approach is correct if we assume that the experimental data contains enough information about <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span> </li>\n</ol>\n<p>The first case is of little interest to us. In the second case, we get the following formula for the solution:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mrow><mo fence=\"true\">⟨</mo><msub><mi>φ</mi><mi>i</mi></msub><mo fence=\"true\">⟩</mo></mrow><mo>=</mo><mfrac><mrow><mo>∫</mo><mi>d</mi><mi>φ</mi><mtext>&ThinSpace;</mtext><msub><mi>φ</mi><mi>i</mi></msub><mi>P</mi><mo>(</mo><mi>f</mi><mi mathvariant=\"normal\">∣</mi><mi>φ</mi><mo>)</mo><mo>∫</mo><mi>d</mi><mi>α</mi><mtext>&ThinSpace;</mtext><mi>P</mi><mo>(</mo><mi>α</mi><mo>)</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow><mrow><mo>∫</mo><mi>d</mi><mi>φ</mi><mi>P</mi><mo>(</mo><mi>f</mi><mi mathvariant=\"normal\">∣</mi><mi>φ</mi><mo>)</mo><mo>∫</mo><mi>d</mi><mi>α</mi><mtext>&ThinSpace;</mtext><mi>P</mi><mo>(</mo><mi>α</mi><mo>)</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mo>(</mo><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">Ω</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>)</mo><mo>)</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">\\left\\langle \\varphi_i \\right\\rangle = \\frac{\\int d\\varphi\\, \\varphi_i P(f|\\varphi) \\int\\limits d\\alpha\\,P(\\alpha)  \\alpha^{\\frac{Rg(\\Omega)}{2}} \\exp(-\\frac{\\alpha}{2} (\\vec{\\varphi},\\Omega\\vec{\\varphi}))}{\\int d\\varphi P(f|\\varphi) \\int\\limits d\\alpha\\,P(\\alpha)  \\alpha^{\\frac{Rg(\\Omega)}{2}} \\exp(-\\frac{\\alpha}{2} (\\vec{\\varphi},\\Omega\\vec{\\varphi}))}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\">⟨</span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose delimcenter\" style=\"top:0em;\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.13445em;vertical-align:-1.31em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.8244500000000001em;\"><span style=\"top:-2.1244500000000004em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">φ</span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord\">∣</span><span class=\"mord mathdefault\">φ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.075em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.695392em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span><span style=\"top:-3.3194500000000002em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.8244500000000006em;\"><span class=\"pstrut\" style=\"height:3.0894500000000003em;\"></span><span class=\"mord\"><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\">φ</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord\">∣</span><span class=\"mord mathdefault\">φ</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;\">∫</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\">d</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.08945em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.695392em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.0037em;\">α</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">Ω</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.31em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>\n<p>The third case will be considered in the next section using the example of Gaussian noises in an experiment.</p>\n<h3>Gaussian noises case</h3>\n<p>The case where the errors in the experiment are Gaussian distributed is remarkable in that an analytical solution to our problem can be obtained. The solution and its error will be as follows:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover><mo>=</mo><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><msup><mi>α</mi><mo>∗</mo></msup><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><msup><mn>1</mn><mi>T</mi></msup></mrow></msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\vec{\\varphi}  = (K^T\\Sigma^{-1}K +\\alpha^*\\Omega)^{-1}K^T\\Sigma^{-1^{T}}\\vec{f}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\">φ</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1413309999999999em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.306365em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.738696em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.056365em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9190928571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msub><mi mathvariant=\"normal\">Σ</mi><mover accent=\"true\"><mi>φ</mi><mo>⃗</mo></mover></msub><mo>=</mo><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><msup><mi>α</mi><mo>∗</mo></msup><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding=\"application/x-tex\">\\Sigma_{\\vec{\\varphi}}  = (K^T\\Sigma^{-1}K+\\alpha^*\\Omega)^{-1}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9744379999999999em;vertical-align:-0.2911079999999999em;\"></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.5450000000000004em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord accent mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">φ</span></span></span><span style=\"top:-2.714em;\"><span class=\"pstrut\" style=\"height:2.714em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay mtight\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2911079999999999em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1413309999999999em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.1141079999999999em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.738696em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span></span></span></span></span>\n<p>where <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi mathvariant=\"normal\">Σ</mi></mrow><annotation encoding=\"application/x-tex\">\\Sigma</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord\">Σ</span></span></span></span> - covariance matrix of a multidimensional Gaussian distribution, <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><msup><mi>α</mi><mo>∗</mo></msup></mrow><annotation encoding=\"application/x-tex\">\\alpha^*</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.688696em;vertical-align:0em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.688696em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span></span></span></span></span></span></span></span> - the most probable value of the parameter <span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding=\"application/x-tex\">\\alpha</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span></span></span></span>, which is determined from the condition of maximum a posteriori probability density:</p>\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math><semantics><mrow><mi>P</mi><mo>(</mo><mi>α</mi><mi mathvariant=\"normal\">∣</mi><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo><mo>=</mo><mi>C</mi><mn>39</mn><mo separator=\"true\">;</mo><msup><mi>α</mi><mfrac><mrow><mi>R</mi><mi>g</mi><mo>(</mo><mi mathvariant=\"normal\">Ω</mi><mo>)</mo></mrow><mn>2</mn></mfrac></msup><msqrt><mrow><mi mathvariant=\"normal\">∣</mi><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant=\"normal\">∣</mi></mrow></msqrt><mi>exp</mi><mo>⁡</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><mo>(</mo><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>K</mi><mo>+</mo><mi>α</mi><mi mathvariant=\"normal\">Ω</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>K</mi><mi>T</mi></msup><msup><mi mathvariant=\"normal\">Σ</mi><mrow><mo>−</mo><msup><mn>1</mn><mi>T</mi></msup></mrow></msup><mover accent=\"true\"><mi>f</mi><mo>⃗</mo></mover><mo>)</mo></mrow><annotation encoding=\"application/x-tex\">P(\\alpha|\\vec{f}) = C39; \\alpha^{\\frac{Rg(\\Omega)}{2}}\\sqrt{|(K^T\\Sigma^{-1}K+\\alpha\\Omega)^{-1}|}\\exp(\\frac{1}{2} \\vec{f}^T\\Sigma^{-1}K^{T}(K^T\\Sigma^{-1}K+\\alpha\\Omega)^{-1}K^T\\Sigma^{-1^{T}}\\vec{f})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.2274399999999999em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">∣</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">C</span><span class=\"mord\">3</span><span class=\"mord\">9</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.13945em;\"><span style=\"top:-3.4130000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0377857142857143em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span><span style=\"top:-3.5020714285714285em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.00773em;\">R</span><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">Ω</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"></span></span></span></span></span></span></span></span></span></span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2925405em;\"><span class=\"svg-align\" style=\"top:-3.8em;\"><span class=\"pstrut\" style=\"height:3.8em;\"></span><span class=\"mord\" style=\"padding-left:1em;\"><span class=\"mord\">∣</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.767331em;\"><span style=\"top:-2.9890000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.740108em;\"><span style=\"top:-2.989em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\">∣</span></span></span><span style=\"top:-3.2525405em;\"><span class=\"pstrut\" style=\"height:3.8em;\"></span><span class=\"hide-tail\" style=\"min-width:1.02em;height:1.8800000000000001em;\"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,\n572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,\n-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39\nc-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60\ns208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,\n-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10z\nM1001 80H400000v40H1013z'/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5474595em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">exp</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">2</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.306365em;vertical-align:-0.25em;\"></span><span class=\"mord mathdefault\" style=\"margin-right:0.0037em;\">α</span><span class=\"mord\">Ω</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.864108em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913309999999999em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord\">Σ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.056365em;\"><span style=\"top:-3.113em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9190928571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.06882999999999997em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>\n<p>As an example, we consider the reconstruction of a spectrum consisting of two Gaussian peaks that fell under the action of an integral step kernel (Heaviside function).</p>\n<img src=\"/images/projects/math/deconvolution.png\" alt=\"deconvolution\"/>","frontmatter":{"shortTitle":"Inverse problems","title":"Statistical regularization of incorrect inverse problems","id":"deconvolution"}}},{"node":{"html":"<table>\n    <tbody><tr><td>\n        <div classname=\"col-lg-9\">\n            <img src=\"/images/projects/math/gears_animated.gif\" alt=\"Under construction...\">\n        </div>\n    </td>\n    <td>\n        <div classname=\"col-lg-8\" align=\"center\"><h3>This section is being finalized ...</h3></div>\n    </td>\n</tr></tbody></table>","frontmatter":{"shortTitle":"Significance functions","title":"Optimal experiment planning with parameter significance functions","id":"significance"}}}]}},"pageContext":{"isCreatedByStatefulCreatePages":true,"intl":{"language":"ru","languages":["ru","en"],"messages":{"title":"NPM Group","language":"ru","description":"Лаборатория методов ядерной физики","header.news":"Новости","header.group":"Группа","header.projects":"Проекты","header.partners":"Партнёры","notfound.header":"404: НЕ НАЙДЕНО","notfound.description":"Вы перешли по несуществующему пути","jumbotron.labintro":"Лаборатория методов ядерно-физических экспериментов","jumbotron.lead":"Особенности нашего подхода к решению научных задач сегодняшнего времени: ","jumbotron.list":"<ul><li>Лаборатория была создана на базе МФТИ, что позволяет привлекать большое количество заинтересованных лиц из числа студентов.</li><li>Благодаря совмещению научной работы с образовательным процессом мы обеспечиваем преемственность научного опыта.</li><li>Структура нашей лаборатории позволяет принимать участие в экспериментах мирового уровня даже студентам младших курсов.</li><li>Мы применяем самые современные методы в работе на физических экспериментах.</li></ul><p />","jumbotron.about":"О нашей лаборатории","more.nuclear_title":"Ядерная физика","more.nuclear_body":"Лаборатория принимает участие в нескольких международных экспериментах в области физики частиц, таких как эксперимент по безнейтринному двойному бета-распаду GERDA, эксперимент по поиску массы нейтрино Троицк ню-масс и так далее.","more.nuclear_more":"Подробнее »","more.education_title":"Образование","more.education_body":"В задачи лаборатории входит разработка новых образовательных программ по физике и методике проведения физического эксперимента, а также совершенствование существующей методической и информационной базы в МФТИ и академических институтах.","more.education_more":"Подробнее »","more.software_title":"Компьютерные методы","more.software_body":"Одним из основных направлений деятельности является разработка вычислительных методов и открытого программного обеспечения для использования в образовании и научной деятельности.","more.software_more":"Подробнее »","more.news":"Последние новости","about.title":"Группа методики ядерно-физического эксперимента","about.descr":"Группа была создана в 2015 году на базе кафедры общей физики МФТИ, нескольких лабораторий ИЯИ РАН и при поддержке лаборатории физики высоких энергий МФТИ. Цель создания - разработка методов для проведения и анализа данных экспериментов в области физики частиц и ядерной физики. Помимо этого участники группы занимаются внедрением современных информационных технологий в экспериментальную физику и образование.","about.pubs.title":"Публикации","about.pubs.available1":"Публикации группы доступны на ","about.pubs.available2":"отдельной странице","about.contacts.title":"Контактная информация","about.contacts.mail":"Электронный адрес: ","about.contacts.telegram":"Телеграм канал: ","partners.mipt.title_fund":"Кафедра общей физики МФТИ","partners.mipt.description_fund":"Кафедра общей физики является основной точкой соприкосновения для ученых и преподавателей с одной стороны и студентов с другой стороны. Тесное сотрудничество с кафдерой является залогом постоянного притока молодых сотрудников, а также постоянного самосовершенствования членов группы, работающих со студентами.","partners.mipt.title_energy":"Лаборатория физики высоких энергий МФТИ","partners.mipt.description_energy":"Тесное сотрудничество с лабораторией физики высоких энергий позволяет осуществлять прямой контакт между образованием и научным сообществом, не выходя за рамки МФТИ.","partners.jb.description":"Лаборатория активно сотрудничает с компанией JetBrains во внедрении языка Kotlin в научном программировании, преподавании Kotlin и разработке библиотек на Kotlin.","partners.jbr.description":"Группа разработки программного обеспечения входит в международное научное объединение JetBrains Research.","partners.ras.title_exp":"Отдел экспериментальной физики ИЯИ РАН","partners.ras.description_exp":"Ведется очень плотное сотруднничество с ОЭФ ИЯИ РАН в рамках коллабораций Troitsk nu-mass и KATRIN, а также в плане подготовки квалифицированных кадров для работы на эксперименте NICA и в других ускорительных экспериментах. В рамках сотрудничества реализуются как научные так и образовательные задачи.","partners.ras.title_beam":"Лаборатория пучка ИЯИ РАН","partners.ras.description_beam":"Лаборатория пучка линейного ускорителя ИЯИ РАН отвечает за проводку и диагностику пучка ускорителя, а также ведет разработки систем диагностики пучка, используемых по всему миру. Группа ведет несколько совместных образовательных проектов с этой лабораторией.","partners.ras.title_education":"Научно-образовательный центр ИЯИ РАН","partners.ras.description_education":"Часть студентов, участвующих в группе обучается в научно-образовательном центре ИЯИ РАН.","partners.ras.title_iki":"ИКИ РАН","partners.ras.description_iki":"Группа участвует в математическом моделировании электрических разрядов в атмосфере.","physics.bc_title":"Физика","physics.title":"Ядерная физика","physics.description":"Традиционно к ядерной физике относят не только исследования, связанные со структурой атомного ядра и ядерными реакциями, но и всю физику элементарных частиц, а также отчасти некоторые разделы астрофизики и космологии. В настоящее время усилия нашей группы сосредоточены в области так называемых неускорительных экспериментов в физике элементарных частиц.","education.bc_title":"Образование","education.title":"Образование","education.description":"Образовательные проекты в побласти ядерной физики и методов проведения и анализа результатов физического эксперимента являются одним из ключевых направлений деятельности группы.","education.course1":"Подробная информация доступна на ","education.course2":"странице курса","math.bc_title":"Математика","math.title":"Математические методы","math.description":"Математическое моделирование физических процессов и математические методы анализа данных являются неотъемлимой частью современной экспериментальной физики. 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