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interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Probabilitat_posterior" title="Probabilitat posterior – Catalan" lang="ca" hreflang="ca" data-title="Probabilitat posterior" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/A-posteriori-Wahrscheinlichkeit" title="A-posteriori-Wahrscheinlichkeit – German" lang="de" hreflang="de" data-title="A-posteriori-Wahrscheinlichkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Aposterioorne_t%C3%B5en%C3%A4osus" title="Aposterioorne tõenäosus – Estonian" lang="et" hreflang="et" data-title="Aposterioorne tõenäosus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Probabilidad_a_posteriori" title="Probabilidad a posteriori – Spanish" lang="es" hreflang="es" data-title="Probabilidad a posteriori" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84_%D9%BE%D8%B3%DB%8C%D9%86" title="احتمال پسین – Persian" lang="fa" hreflang="fa" data-title="احتمال پسین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Probabilit%C3%A9_a_posteriori" title="Probabilité a posteriori – French" lang="fr" hreflang="fr" data-title="Probabilité a posteriori" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%ED%9B%84_%ED%99%95%EB%A5%A0" title="사후 확률 – Korean" lang="ko" hreflang="ko" data-title="사후 확률" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Probabilit%C3%A0_a_posteriori" title="Probabilità a posteriori – Italian" lang="it" hreflang="it" data-title="Probabilità a posteriori" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A1%D7%AA%D7%91%D7%A8%D7%95%D7%AA_%D7%A4%D7%95%D7%A1%D7%98%D7%A8%D7%99%D7%95%D7%A8%D7%99%D7%AA" title="הסתברות פוסטריורית – Hebrew" lang="he" hreflang="he" data-title="הסתברות פוסטריורית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BF%D0%BE%D1%81%D1%82%D0%B5%D1%80%D0%B8%D0%BE%D1%80%D0%BB%D1%8B%D2%9B_%D1%8B%D2%9B%D1%82%D0%B8%D0%BC%D0%B0%D0%BB%D0%B4%D1%8B%D2%9B" title="Апостериорлық ықтималдық – Kazakh" lang="kk" hreflang="kk" data-title="Апостериорлық ықтималдық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8B%E5%BE%8C%E7%A2%BA%E7%8E%87" title="事後確率 – Japanese" lang="ja" hreflang="ja" data-title="事後確率" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawdopodobie%C5%84stwo_a_posteriori" title="Prawdopodobieństwo a posteriori – Polish" lang="pl" hreflang="pl" data-title="Prawdopodobieństwo a posteriori" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Probabilidade_a_posteriori" title="Probabilidade a posteriori – Portuguese" lang="pt" hreflang="pt" data-title="Probabilidade a posteriori" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BF%D0%BE%D1%81%D1%82%D0%B5%D1%80%D0%B8%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Апостериорная вероятность – Russian" lang="ru" hreflang="ru" data-title="Апостериорная вероятность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Probabiliteti_i_pas%C3%ABm" title="Probabiliteti i pasëm – Albanian" lang="sq" hreflang="sq" data-title="Probabiliteti i pasëm" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BF%D0%BE%D1%81%D1%82%D0%B5%D1%80%D1%96%D0%BE%D1%80%D0%BD%D0%B0_%D0%B9%D0%BC%D0%BE%D0%B2%D1%96%D1%80%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Апостеріорна ймовірність – Ukrainian" lang="uk" hreflang="uk" data-title="Апостеріорна ймовірність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/X%C3%A1c_su%E1%BA%A5t_h%E1%BA%ADu_nghi%E1%BB%87m" title="Xác suất hậu nghiệm – Vietnamese" lang="vi" hreflang="vi" data-title="Xác suất hậu nghiệm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BE%8C%E9%A9%97%E6%A6%82%E7%8E%87" title="後驗概率 – Cantonese" lang="yue" hreflang="yue" 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Bayes_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/80px-Bayes_icon.svg.png" decoding="async" width="80" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/120px-Bayes_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/160px-Bayes_icon.svg.png 2x" data-file-width="500" data-file-height="300" /></a></span></td></tr><tr><td class="sidebar-content"> <a class="mw-selflink selflink">Posterior</a> = <a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> × <a href="/wiki/Prior_probability" title="Prior probability">Prior</a> ÷ <a href="/wiki/Marginal_likelihood" title="Marginal likelihood">Evidence</a></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li> <li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a></li> <li><a href="/wiki/Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">Coherence</a></li> <li><a href="/wiki/Cox%27s_theorem" title="Cox's theorem">Cox's theorem</a></li> <li><a href="/wiki/Cromwell%27s_rule" title="Cromwell's rule">Cromwell's rule</a></li> <li><a href="/wiki/Likelihood_principle" title="Likelihood principle">Likelihood principle</a></li> <li><a href="/wiki/Principle_of_indifference" title="Principle of indifference">Principle of indifference</a></li> <li><a href="/wiki/Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Model building</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Conjugate_prior" title="Conjugate prior">Conjugate prior</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Linear regression</a></li> <li><a href="/wiki/Empirical_Bayes_method" title="Empirical Bayes method">Empirical Bayes</a></li> <li><a href="/wiki/Bayesian_hierarchical_modeling" title="Bayesian hierarchical modeling">Hierarchical model</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Posterior approximation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a></li> <li><a href="/wiki/Laplace%27s_approximation" title="Laplace's approximation">Laplace's approximation</a></li> <li><a href="/wiki/Integrated_nested_Laplace_approximations" title="Integrated nested Laplace approximations">Integrated nested Laplace approximations</a></li> <li><a href="/wiki/Variational_Bayesian_methods" title="Variational Bayesian methods">Variational inference</a></li> <li><a href="/wiki/Approximate_Bayesian_computation" title="Approximate Bayesian computation">Approximate Bayesian computation</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimators</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayesian_estimator" class="mw-redirect" title="Bayesian estimator">Bayesian estimator</a></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum a posteriori estimation</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Evidence approximation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Evidence_lower_bound" title="Evidence lower bound">Evidence lower bound</a></li> <li><a href="/wiki/Nested_sampling_algorithm" title="Nested sampling algorithm">Nested sampling</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Model evaluation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a> (<a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">Schwarz criterion</a>)</li> <li><a href="/wiki/Bayesian_model_averaging" class="mw-redirect" title="Bayesian model averaging">Model averaging</a></li> <li><a href="/wiki/Posterior_predictive_distribution" title="Posterior predictive distribution">Posterior predictive</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bayesian_statistics" title="Template:Bayesian statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bayesian_statistics" title="Template talk:Bayesian statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bayesian_statistics" title="Special:EditPage/Template:Bayesian statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>posterior probability</b> is a type of <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a> that results from <a href="/wiki/Bayesian_updating" class="mw-redirect" title="Bayesian updating">updating</a> the <a href="/wiki/Prior_probability" title="Prior probability">prior probability</a> with information summarized by the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood</a> via an application of <a href="/wiki/Bayes%27_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> From an <a href="/wiki/Bayesian_epistemology" title="Bayesian epistemology">epistemological perspective</a>, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the context of <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, the <b>posterior <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a></b> usually describes the epistemic uncertainty about <a href="/wiki/Statistical_parameter" title="Statistical parameter">statistical parameters</a> conditional on a collection of observed data. From a given posterior distribution, various <a href="/wiki/Point_estimate" class="mw-redirect" title="Point estimate">point</a> and <a href="/wiki/Interval_estimate" class="mw-redirect" title="Interval estimate">interval estimates</a> can be derived, such as the <a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">maximum a posteriori</a> (MAP) or the <a href="/wiki/Highest_posterior_density_interval" class="mw-redirect" title="Highest posterior density interval">highest posterior density interval</a> (HPDI).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_in_the_distributional_case">Definition in the distributional case</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=1" title="Edit section: Definition in the distributional case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Bayesian statistics, the posterior probability is the probability of the parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> given the evidence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, and is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta |X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\theta |X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2594603c1c2b622471d9a19d1ea54daa152026b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.785ex; height:2.843ex;" alt="{\displaystyle p(\theta |X)}"></span>. </p><p>It contrasts with the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a>, which is the probability of the evidence given the parameters: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X|\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(X|\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1665d485e91e5a3e953a937574c78389668777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.785ex; height:2.843ex;" alt="{\displaystyle p(X|\theta )}"></span>. </p><p>The two are related as follows: </p><p>Given a <a href="/wiki/Prior_probability" title="Prior probability">prior</a> belief that a <a href="/wiki/Probability_density_function" title="Probability density function">probability distribution function</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b66e40cc46404eb7159d0368dab9cfba9fd96471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.159ex; height:2.843ex;" alt="{\displaystyle p(\theta )}"></span> and that the observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> have a likelihood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x|\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x|\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65079ab6195ae61ca993e8f25b261fbfe7b45dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.135ex; height:2.843ex;" alt="{\displaystyle p(x|\theta )}"></span>, then the posterior probability is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta |x)={\frac {p(x|\theta )}{p(x)}}p(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\theta |x)={\frac {p(x|\theta )}{p(x)}}p(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d499ca6e5b854fe444b3c08da3f19487589009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:20.185ex; height:6.509ex;" alt="{\displaystyle p(\theta |x)={\frac {p(x|\theta )}{p(x)}}p(\theta )}"></span>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span> is the normalizing constant and is calculated as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=\int p(x|\theta )p(\theta )d\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=\int p(x|\theta )p(\theta )d\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16eff4b7bf72a545eb138aae81e0bee2c614c9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:22.498ex; height:5.676ex;" alt="{\displaystyle p(x)=\int p(x|\theta )p(\theta )d\theta }"></span></dd></dl> <p>for continuous <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>, or by summing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x|\theta )p(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x|\theta )p(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce3eb184426f57c7ab0bec6c6cbba2feddb4020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.204ex; height:2.843ex;" alt="{\displaystyle p(x|\theta )p(\theta )}"></span> over all possible values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> for discrete <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>.<sup id="cite_ref-BDA_7-0" class="reference"><a href="#cite_note-BDA-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The posterior probability is therefore <a href="/wiki/Direct_proportionality" class="mw-redirect" title="Direct proportionality">proportional to</a> the product <i>Likelihood · Prior probability</i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose there is a school with 60% boys and 40% girls as students. The girls wear trousers or skirts in equal numbers; all boys wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem. </p><p>The event <span class="texhtml mvar" style="font-style:italic;">G</span> is that the student observed is a girl, and the event <span class="texhtml mvar" style="font-style:italic;">T</span> is that the student observed is wearing trousers. To compute the posterior probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(G|T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(G|T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507faeeeb1d119c9c3c7c918372786b789f475a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.665ex; height:2.843ex;" alt="{\displaystyle P(G|T)}"></span>, we first need to know: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a475bb06a0bba3304f724843f91c64a918fa8a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.381ex; height:2.843ex;" alt="{\displaystyle P(G)}"></span>, or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e593d180a26fd68657ea50368dbfe1a661e652aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.319ex; height:2.843ex;" alt="{\displaystyle P(B)}"></span>, or the probability that the student is not a girl (i.e. a boy) regardless of any other information (<span class="texhtml mvar" style="font-style:italic;">B</span> is the complementary event to <span class="texhtml mvar" style="font-style:italic;">G</span>). This is 60%, or 0.6.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T|G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T|G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd2cedf7c3deb72ec49c66dd79b1ab83b44a5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.665ex; height:2.843ex;" alt="{\displaystyle P(T|G)}"></span>, or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T|B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T|B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069a325e025dd5e7fc2a83b8c9bbd0fa12b6c7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.602ex; height:2.843ex;" alt="{\displaystyle P(T|B)}"></span>, or the probability of the student wearing trousers given that the student is a boy. This is given as 1.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe23eeabf3ea0fa52aee5b90d31d972acd9dc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.191ex; height:2.843ex;" alt="{\displaystyle P(T)}"></span>, or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T)=P(T|G)P(G)+P(T|B)P(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T)=P(T|G)P(G)+P(T|B)P(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5404dd0f3dfb1640ec0e1d6e21e74e95c7b76503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.096ex; height:2.843ex;" alt="{\displaystyle P(T)=P(T|G)P(G)+P(T|B)P(B)}"></span> (via the <a href="/wiki/Law_of_total_probability" title="Law of total probability">law of total probability</a>), this is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(T)=0.5\times 0.4+1\times 0.6=0.8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.5</mn> <mo>×</mo> <mn>0.4</mn> <mo>+</mo> <mn>1</mn> <mo>×</mo> <mn>0.6</mn> <mo>=</mo> <mn>0.8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(T)=0.5\times 0.4+1\times 0.6=0.8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcadbaee6226dcf94bb85463c8b7cae1c4042d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.959ex; height:2.843ex;" alt="{\displaystyle P(T)=0.5\times 0.4+1\times 0.6=0.8}"></span>.</li></ul> <p>Given all this information, the <b>posterior probability</b> of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(G|T)={\frac {P(T|G)P(G)}{P(T)}}={\frac {0.5\times 0.4}{0.8}}=0.25.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>0.5</mn> <mo>×</mo> <mn>0.4</mn> </mrow> <mn>0.8</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.25.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(G|T)={\frac {P(T|G)P(G)}{P(T)}}={\frac {0.5\times 0.4}{0.8}}=0.25.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d600d0664fbf0c64aefcfffb76cf616a68b919d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.243ex; height:6.509ex;" alt="{\displaystyle P(G|T)={\frac {P(T|G)P(G)}{P(T)}}={\frac {0.5\times 0.4}{0.8}}=0.25.}"></span></dd></dl> <p>An intuitive way to solve this is to assume the school has <i>N</i> students. Number of boys = 0.6<i>N</i> and number of girls = 0.4<i>N</i>. If <i>N</i> is sufficiently large, total number of trouser wearers = 0.6<i>N</i> + 50% of 0.4<i>N</i>. And number of girl trouser wearers = 50% of 0.4<i>N</i>. Therefore, in the population of trousers, girls are (50% of 0.4<i>N</i>)/(0.6<i>N</i> + 50% of 0.4<i>N</i>) = 25%. In other words, if you separated out the group of trouser wearers, a quarter of that group will be girls. Therefore, if you see trousers, the most you can deduce is that you are looking at a single sample from a subset of students where 25% are girls. And by definition, chance of this random student being a girl is 25%. Every Bayes-theorem problem can be solved in this way.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=3" title="Edit section: Calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The posterior probability distribution of one <a href="/wiki/Random_variable" title="Random variable">random variable</a> given the value of another can be calculated with <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a> by multiplying the <a href="/wiki/Prior_probability_distribution" class="mw-redirect" title="Prior probability distribution">prior probability distribution</a> by the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a>, and then dividing by the <a href="/wiki/Normalizing_constant" title="Normalizing constant">normalizing constant</a>, as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X\mid Y=y}(x)={f_{X}(x){\mathcal {L}}_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X\mid Y=y}(x)={f_{X}(x){\mathcal {L}}_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d6566d3e3d2fa637cccceac0d1fbea40961914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.316ex; height:7.176ex;" alt="{\displaystyle f_{X\mid Y=y}(x)={f_{X}(x){\mathcal {L}}_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}}}"></span></dd></dl> <p>gives the posterior <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> for a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> given the data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678864c5e9a7ce08acfc22d0d7f726d2cade5b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.027ex; height:2.509ex;" alt="{\displaystyle Y=y}"></span>, where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29310a010e7f7dfa33ba69bcf1ef9ec166d461dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)}"></span> is the prior density of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc62084cf308fe3c8d9792f3835385d0e7452f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.947ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)}"></span> is the likelihood function as a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75045952bda2a45b9c73ca049b5ef57647d1486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.859ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}"></span> is the normalizing constant, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X\mid Y=y}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X\mid Y=y}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7fa5c5e9bbc94ff5f447fe459c5389ffe1eee8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.717ex; height:3.176ex;" alt="{\displaystyle f_{X\mid Y=y}(x)}"></span> is the posterior density of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> given the data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678864c5e9a7ce08acfc22d0d7f726d2cade5b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.027ex; height:2.509ex;" alt="{\displaystyle Y=y}"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Credible_interval">Credible interval</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=4" title="Edit section: Credible interval"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Posterior probability is a conditional probability conditioned on randomly observed data. Hence it is a random variable. For a random variable, it is important to summarize its amount of uncertainty. One way to achieve this goal is to provide a <a href="/wiki/Credible_interval" title="Credible interval">credible interval</a> of the posterior probability.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=5" title="Edit section: Classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a>, posterior probabilities reflect the uncertainty of assessing an observation to particular class, see also <a href="/wiki/Class-membership_probabilities" class="mw-redirect" title="Class-membership probabilities">class-membership probabilities</a>. While <a href="/wiki/Statistical_classification" title="Statistical classification">statistical classification</a> methods by definition generate posterior probabilities, Machine Learners usually supply membership values which do not induce any probabilistic confidence. It is desirable to transform or rescale membership values to class-membership probabilities, since they are comparable and additionally more easily applicable for post-processing.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a></li> <li><a href="/wiki/Probability_of_success" title="Probability of success">Probability of success</a></li> <li><a href="/wiki/Bayesian_epistemology" title="Bayesian epistemology">Bayesian epistemology</a></li> <li><a href="/wiki/Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis–Hastings algorithm</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLambert2018" class="citation book cs1">Lambert, Ben (2018). 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James (1989). "Approximations, Numerical Methods, and Computer Programs". <i>Bayesian Statistics : Principles, Models, and Applications</i>. New York: John Wiley & Sons. pp. <span class="nowrap">69–</span>102. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-63729-7" title="Special:BookSources/0-471-63729-7"><bdi>0-471-63729-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Approximations%2C+Numerical+Methods%2C+and+Computer+Programs&rft.btitle=Bayesian+Statistics+%3A+Principles%2C+Models%2C+and+Applications&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E69-%3C%2Fspan%3E102&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=0-471-63729-7&rft.aulast=Press&rft.aufirst=S.+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristopher_M._Bishop2006" class="citation book cs1">Christopher M. 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Retrieved <span class="nowrap">2022-08-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=formulasearchengine.com&rft.atitle=Posterior+probability+-+formulasearchengine&rft_id=https%3A%2F%2Fformulasearchengine.com%2Fwiki%2FPosterior_probability&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClydeÇetinkaya-RundelRundelBanks" class="citation book cs1">Clyde, Merlise; Çetinkaya-Rundel, Mine; Rundel, Colin; Banks, David; Chai, Christine; Huang, Lizzy. <a rel="nofollow" class="external text" href="https://statswithr.github.io/book/the-basics-of-bayesian-statistics.html"><i>Chapter 1 The Basics of Bayesian Statistics | An Introduction to Bayesian Thinking</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chapter+1+The+Basics+of+Bayesian+Statistics+%7C+An+Introduction+to+Bayesian+Thinking&rft.aulast=Clyde&rft.aufirst=Merlise&rft.au=%C3%87etinkaya-Rundel%2C+Mine&rft.au=Rundel%2C+Colin&rft.au=Banks%2C+David&rft.au=Chai%2C+Christine&rft.au=Huang%2C+Lizzy&rft_id=https%3A%2F%2Fstatswithr.github.io%2Fbook%2Fthe-basics-of-bayesian-statistics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoedekerKearns2019" class="citation journal cs1">Boedeker, Peter; Kearns, Nathan T. (2019-07-09). <a rel="nofollow" class="external text" href="http://journals.sagepub.com/doi/10.1177/2515245919849378">"Linear Discriminant Analysis for Prediction of Group Membership: A User-Friendly Primer"</a>. <i>Advances in Methods and Practices in Psychological Science</i>. <b>2</b> (3): <span class="nowrap">250–</span>263. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F2515245919849378">10.1177/2515245919849378</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2515-2459">2515-2459</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:199007973">199007973</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Methods+and+Practices+in+Psychological+Science&rft.atitle=Linear+Discriminant+Analysis+for+Prediction+of+Group+Membership%3A+A+User-Friendly+Primer&rft.volume=2&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E250-%3C%2Fspan%3E263&rft.date=2019-07-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199007973%23id-name%3DS2CID&rft.issn=2515-2459&rft_id=info%3Adoi%2F10.1177%2F2515245919849378&rft.aulast=Boedeker&rft.aufirst=Peter&rft.au=Kearns%2C+Nathan+T.&rft_id=http%3A%2F%2Fjournals.sagepub.com%2Fdoi%2F10.1177%2F2515245919849378&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Posterior_probability&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLancaster2004" class="citation book cs1">Lancaster, Tony (2004). <i>An Introduction to Modern Bayesian Econometrics</i>. Oxford: Blackwell. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4051-1720-6" title="Special:BookSources/1-4051-1720-6"><bdi>1-4051-1720-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Modern+Bayesian+Econometrics&rft.place=Oxford&rft.pub=Blackwell&rft.date=2004&rft.isbn=1-4051-1720-6&rft.aulast=Lancaster&rft.aufirst=Tony&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2004" class="citation book cs1">Lee, Peter M. (2004). <i>Bayesian Statistics : An Introduction</i> (3rd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-340-81405-5" title="Special:BookSources/0-340-81405-5"><bdi>0-340-81405-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Statistics+%3A+An+Introduction&rft.edition=3rd&rft.pub=Wiley&rft.date=2004&rft.isbn=0-340-81405-5&rft.aulast=Lee&rft.aufirst=Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APosterior+probability" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Posterior_probability&oldid=1274985650">https://en.wikipedia.org/w/index.php?title=Posterior_probability&oldid=1274985650</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Bayesian_statistics" title="Category:Bayesian statistics">Bayesian statistics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 February 2025, at 12:33<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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