Bayesian hierarchical modeling

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#2-stage_hierarchical_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>2-stage hierarchical model</span> </div> </a> <ul id="toc-2-stage_hierarchical_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3-stage_hierarchical_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#3-stage_hierarchical_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>3-stage hierarchical model</span> </div> </a> <ul id="toc-3-stage_hierarchical_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bayesian_nonlinear_mixed-effects_model" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bayesian_nonlinear_mixed-effects_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bayesian nonlinear 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.sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .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 href="/wiki/Posterior_probability" title="Posterior probability">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 class="mw-selflink selflink">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><b>Bayesian hierarchical modelling</b> is a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> written in multiple levels (hierarchical form) that estimates the <a href="/wiki/Parametric_model" title="Parametric model">parameters</a> of the <a href="/wiki/Posterior_probability" title="Posterior probability">posterior distribution</a> using the <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian method</a>.<sup id="cite_ref-allenby_1-0" class="reference"><a href="#cite_note-allenby-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The sub-models combine to form the hierarchical model, and <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a> is used to integrate them with the observed data and account for all the uncertainty that is present. The result of this integration is the posterior distribution, also known as the updated probability estimate, as additional evidence on the <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a> is acquired. </p><p><a href="/wiki/Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">Frequentist statistics</a> may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as <a href="/wiki/Random_variable" title="Random variable">random variables</a> and its use of subjective information in establishing assumptions on these parameters.<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> As the approaches answer different questions the formal results aren't technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be <a href="/wiki/Robust_statistics" title="Robust statistics">robust</a>, with the posterior distribution less sensitive to the more flexible hierarchical priors. </p><p>Hierarchical modeling is used when information is available on several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own temporal profile of daily infected cases.<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> In <a href="/wiki/Decline_curve_analysis" title="Decline curve analysis">decline curve analysis</a> to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own temporal profile of oil or gas production rates (usually, barrels per month).<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> Data structure for the hierarchical modeling retains nested data structure. The hierarchical form of analysis and organization helps in the understanding of multiparameter problems and also plays an important role in developing computational strategies.<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin20046_5-0" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin20046-5"><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="Philosophy">Philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=1" title="Edit section: Philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters.<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin2004117_6-0" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin2004117-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Individual degrees of belief, expressed in the form of probabilities, come with uncertainty.<sup id="cite_ref-good_7-0" class="reference"><a href="#cite_note-good-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Amidst this is the change of the degrees of belief over time. As was stated by Professor <a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">José M. Bernardo</a> and Professor <a href="/wiki/Adrian_Smith_(statistician)" title="Adrian Smith (statistician)">Adrian F. Smith</a>, “The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality.” These subjective probabilities are more directly involved in the mind rather than the physical probabilities.<sup id="cite_ref-good_7-1" class="reference"><a href="#cite_note-good-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event.<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="Bayes'_theorem"><span id="Bayes.27_theorem"></span>Bayes' theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=2" title="Edit section: Bayes' theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options.<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin20046–8_9-0" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin20046–8-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital <i>j</i> having survival 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 \theta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span>, the survival probability will be updated with the occurrence of <i>y</i>, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. </p><p>In order to make updated probability statements about <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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span>, given the occurrence of event <i>y</i>, we must begin with a model providing a <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> for <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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span> and <i>y</i>. This can be written as a product of the two distributions that are often referred to as the prior distribution <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/543ecac78af032493892f6608bff05542e961edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle P(\theta )}"></span> and the <a href="/wiki/Sampling_distribution" title="Sampling distribution">sampling distribution</a> <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(y\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9202bf7efaffba283ccfb1d67a803c5aebc216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.738ex; height:2.843ex;" alt="{\displaystyle P(y\mid \theta )}"></span> respectively: </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 ,y)=P(\theta )P(y\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7e22232e3ca59645fe02402423e63a6d441c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.316ex; height:2.843ex;" alt="{\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )}"></span></dd></dl> <p>Using the basic property of <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a>, the posterior distribution will yield: </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 \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f481b8cddcc78001d0b1e21e6b1bc7ed898616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.825ex; height:6.509ex;" alt="{\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}}"></span></dd></dl> <p>This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to incorporate the updated belief, <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 \mid y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta \mid y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbdb4a652f3bd3294f621a4d8f43b4264ca9c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.738ex; height:2.843ex;" alt="{\displaystyle P(\theta \mid y)}"></span>, in appropriate and solvable ways.<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin20046–8_9-1" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin20046–8-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Exchangeability">Exchangeability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=3" title="Edit section: Exchangeability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The usual starting point of a statistical analysis is the assumption that the <i>n</i> values <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_{1},y_{2},\ldots ,y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1},y_{2},\ldots ,y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3caca8a9c9ae5b264a0bf5e6d9058c5915d4051c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.957ex; height:2.009ex;" alt="{\displaystyle y_{1},y_{2},\ldots ,y_{n}}"></span> are exchangeable. If no information – other than data <i>y</i> – is available to distinguish any of the <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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span>’s from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry among the parameters in their prior distribution.<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> This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as <a href="/wiki/Independently_and_identically_distributed" class="mw-redirect" title="Independently and identically distributed">independently and identically distributed</a>, given some unknown parameter vector <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>, with distribution <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/543ecac78af032493892f6608bff05542e961edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle P(\theta )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Finite_exchangeability">Finite exchangeability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=4" title="Edit section: Finite exchangeability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a fixed number <i>n</i>, the set <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_{1},y_{2},\ldots ,y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1},y_{2},\ldots ,y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3caca8a9c9ae5b264a0bf5e6d9058c5915d4051c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.957ex; height:2.009ex;" alt="{\displaystyle y_{1},y_{2},\ldots ,y_{n}}"></span> is exchangeable if the joint 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(y_{1},y_{2},\ldots ,y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y_{1},y_{2},\ldots ,y_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7824f1289d07ebe0bd353330b2c4640f8221ac58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.512ex; height:2.843ex;" alt="{\displaystyle P(y_{1},y_{2},\ldots ,y_{n})}"></span> is invariant under <a href="/wiki/Permutation" title="Permutation">permutations</a> of the indices. That is, for every permutation <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> or <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 (\pi _{1},\pi _{2},\ldots ,\pi _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f96cbc21ed810e6412427a2489104bf6421456e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.324ex; height:2.843ex;" alt="{\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})}"></span> of (1, 2, …, <i>n</i>), <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(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d374d3f27debc4dc3c908d53f40b6ce9555243d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.577ex; height:3.009ex;" alt="{\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).}"></span><sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin2004121–125_11-0" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin2004121–125-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with 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 {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"></span> of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the <i>n</i> balls, there will be <i>n</i> − 1 remaining balls left for the next draw. </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 {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Let </mtext> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if the </mtext> </mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th ball is red</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cbcf3f16dbc8ab1ac285f08f5497db1e825d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.532ex; height:6.176ex;" alt="{\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}}"></span></dd></dl> <p>Since the probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second draw, both of which are equal to 1/2 (i.e. <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(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38545851ac08b0e0213b43212d72533a168fc3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.484ex; height:5.176ex;" alt="{\displaystyle [P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}]}"></span>), then <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4db76d658a98219aca14df06d9869d2b43c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}"></span> and <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7377c7399e662562cd420fa5c7ce49cfba574998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{2}}"></span> are exchangeable. </p><p>But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first draw is 0, and is not equal to the probability that the red ball is selected in the second draw which is equal to 1/2 (i.e. <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(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>≠</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71038e75f68a338a7d959d54d3a436bdf166f1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.16ex; height:5.176ex;" alt="{\displaystyle [P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}]}"></span>). Thus, <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4db76d658a98219aca14df06d9869d2b43c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}"></span> and <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7377c7399e662562cd420fa5c7ce49cfba574998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{2}}"></span> are not independent. </p><p>If <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_{1},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}"></span> are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true.<sup id="cite_ref-diaconis_12-0" class="reference"><a href="#cite_note-diaconis-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Infinite_exchangeability">Infinite exchangeability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=5" title="Edit section: Infinite exchangeability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Infinite exchangeability is the property that every finite subset of an infinite sequence <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4db76d658a98219aca14df06d9869d2b43c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}"></span>, <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_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe8188f9b49917164b86fba2f417f2bf7cb1038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.951ex; height:2.009ex;" alt="{\displaystyle y_{2},\ldots }"></span> is exchangeable. That is, for any <i>n</i>, the sequence <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_{1},y_{2},\ldots ,y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1},y_{2},\ldots ,y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3caca8a9c9ae5b264a0bf5e6d9058c5915d4051c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.957ex; height:2.009ex;" alt="{\displaystyle y_{1},y_{2},\ldots ,y_{n}}"></span> is exchangeable.<sup id="cite_ref-diaconis_12-1" class="reference"><a href="#cite_note-diaconis-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Hierarchical_models">Hierarchical models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=6" title="Edit section: Hierarchical models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Components">Components</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=7" title="Edit section: Components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution,<sup id="cite_ref-allenby_1-1" class="reference"><a href="#cite_note-allenby-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> namely: </p> <ol><li><a href="/wiki/Hyperparameter_(Bayesian_statistics)" title="Hyperparameter (Bayesian statistics)">Hyperparameters</a>: parameters of the prior distribution</li> <li><a href="/wiki/Hyperprior" title="Hyperprior">Hyperpriors</a>: distributions of Hyperparameters</li></ol> <p>Suppose a random variable <i>Y</i> follows a normal distribution with parameter <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> as the <a href="/wiki/Mean" title="Mean">mean</a> and 1 as the <a href="/wiki/Variance" title="Variance">variance</a>, that 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 Y\mid \theta \sim N(\theta ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\mid \theta \sim N(\theta ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a8c119a745c8027de44c714211e2ac572175d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.059ex; height:2.843ex;" alt="{\displaystyle Y\mid \theta \sim N(\theta ,1)}"></span>. The <a href="/wiki/Tilde#As_a_relational_operator" title="Tilde">tilde</a> relation <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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter <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> has a distribution given by a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and variance 1, i.e. <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 \mid \mu \sim N(\mu ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>∣</mo> <mi>μ</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \mid \mu \sim N(\mu ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58dbd4ef029a79803909555fada5981b09bc104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.999ex; height:2.843ex;" alt="{\displaystyle \theta \mid \mu \sim N(\mu ,1)}"></span>. Furthermore, <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> follows another distribution given, for example, by the <a href="/wiki/Normal_distribution" title="Normal distribution">standard normal distribution</a>, <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 {\text{N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d75c9c630fc6ab61a159c90be1c07e393aed7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.911ex; height:2.843ex;" alt="{\displaystyle {\text{N}}(0,1)}"></span>. The parameter <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is called the hyperparameter, while its distribution given by <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 {\text{N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d75c9c630fc6ab61a159c90be1c07e393aed7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.911ex; height:2.843ex;" alt="{\displaystyle {\text{N}}(0,1)}"></span> is an example of a hyperprior distribution. The notation of the distribution of <i>Y</i> changes as another parameter is added, i.e. <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\mid \theta ,\mu \sim N(\theta ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo>,</mo> <mi>μ</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\mid \theta ,\mu \sim N(\theta ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9abfa6240328132bb07c4388c1a05fc6af672b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.495ex; height:2.843ex;" alt="{\displaystyle Y\mid \theta ,\mu \sim N(\theta ,1)}"></span>. If there is another stage, say, <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> follows another normal distribution with mean <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> and variance <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span>, meaning <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 \mu \sim N(\beta ,\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \sim N(\beta ,\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516e736d7d56a2807650888499d0de6245a34957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.683ex; height:2.843ex;" alt="{\displaystyle \mu \sim N(\beta ,\epsilon )}"></span>, <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 {\mbox{ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> </mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{ }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/023c16d3f81c9fced45f6235de73d8f3786341bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.581ex; height:0.343ex;" alt="{\displaystyle {\mbox{ }}}"></span><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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> and <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> can also be called hyperparameters while their distributions are hyperprior distributions as well.<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin2004117_6-1" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin2004117-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Framework">Framework</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=8" title="Edit section: Framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8df4e372390588acb968986cfc388e50b930b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.343ex;" alt="{\displaystyle y_{j}}"></span> be an observation and <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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span> a parameter governing the data generating process for <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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8df4e372390588acb968986cfc388e50b930b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.343ex;" alt="{\displaystyle y_{j}}"></span>. Assume further that 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 _{1},\theta _{2},\ldots ,\theta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d55a9c5f3d5f446fcfcd8b73abe3f91618ef00e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.502ex; height:2.843ex;" alt="{\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{j}}"></span> are generated exchangeably from a common population, with distribution governed by a hyperparameter <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>. <br />The Bayesian hierarchical model contains the following stages: </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 {\text{Stage I: }}y_{j}\mid \theta _{j},\phi \sim P(y_{j}\mid \theta _{j},\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Stage I: </mtext> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo>∼</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Stage I: }}y_{j}\mid \theta _{j},\phi \sim P(y_{j}\mid \theta _{j},\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765f37f86fa26bef873048952dccc6e8067b78f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.667ex; height:3.009ex;" alt="{\displaystyle {\text{Stage I: }}y_{j}\mid \theta _{j},\phi \sim P(y_{j}\mid \theta _{j},\phi )}"></span></dd> <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 {\text{Stage II: }}\theta _{j}\mid \phi \sim P(\theta _{j}\mid \phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Stage II: </mtext> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>ϕ</mi> <mo>∼</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Stage II: }}\theta _{j}\mid \phi \sim P(\theta _{j}\mid \phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8c0e1233fd69fa4325c6eacf8462252ed6b00a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.341ex; height:3.009ex;" alt="{\displaystyle {\text{Stage II: }}\theta _{j}\mid \phi \sim P(\theta _{j}\mid \phi )}"></span></dd> <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 {\text{Stage III: }}\phi \sim P(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Stage III: </mtext> </mrow> <mi>ϕ</mi> <mo>∼</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Stage III: }}\phi \sim P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e56b3077b1b3ec867d6a0f2539ba9a3e79b45c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.305ex; height:2.843ex;" alt="{\displaystyle {\text{Stage III: }}\phi \sim P(\phi )}"></span></dd></dl> <p>The likelihood, as seen in stage I 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(y_{j}\mid \theta _{j},\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y_{j}\mid \theta _{j},\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b727bf26aade58556d66a3a96a7b11be6c363df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.96ex; height:3.009ex;" alt="{\displaystyle P(y_{j}\mid \theta _{j},\phi )}"></span>, with <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 _{j},\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta _{j},\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9693985aebc7895508df74d75ff339c3fa66e79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.974ex; height:3.009ex;" alt="{\displaystyle P(\theta _{j},\phi )}"></span> as its prior distribution. Note that the likelihood depends on <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> only through <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 _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd705853e931c59ff5fa4e7149cca4b22019e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.843ex;" alt="{\displaystyle \theta _{j}}"></span>. </p><p>The prior distribution from stage I can be broken down into: </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 _{j},\phi )=P(\theta _{j}\mid \phi )P(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta _{j},\phi )=P(\theta _{j}\mid \phi )P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff3dc004042b24e54f34b39ab0d861e0b3b9a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.89ex; height:3.009ex;" alt="{\displaystyle P(\theta _{j},\phi )=P(\theta _{j}\mid \phi )P(\phi )}"></span> <i>[from the definition of conditional probability]</i></dd></dl> <p>With <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> as its hyperparameter with hyperprior distribution, <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(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf70a07e977c536b7d1446ce59530c237ed35ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.94ex; height:2.843ex;" alt="{\displaystyle P(\phi )}"></span>. </p><p>Thus, the posterior distribution is proportional to: </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(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j},\phi )P(\theta _{j},\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j},\phi )P(\theta _{j},\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae09906afb387914c801fb6474ab35cedf9bde5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.1ex; height:3.009ex;" alt="{\displaystyle P(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j},\phi )P(\theta _{j},\phi )}"></span> <i>[using Bayes' Theorem]</i></dd> <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(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j})P(\theta _{j}\mid \phi )P(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j})P(\theta _{j}\mid \phi )P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48c2189f63e3d188d4c3b8fde54650a2775990f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.524ex; height:3.009ex;" alt="{\displaystyle P(\phi ,\theta _{j}\mid y)\propto P(y_{j}\mid \theta _{j})P(\theta _{j}\mid \phi )P(\phi )}"></span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=9" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To further illustrate this, consider the example: A teacher wants to estimate how well a student did on the <a href="/wiki/SAT" title="SAT">SAT</a>. The teacher uses information on the student’s high school grades and current <a href="/wiki/Grade_point_average" class="mw-redirect" title="Grade point average">grade point average</a> (GPA) to come up with an estimate. The student's current GPA, denoted by <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, has a likelihood given by some probability function with parameter <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>, i.e. <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\mid \theta \sim P(Y\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo>∼</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\mid \theta \sim P(Y\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f7e1a4478ac0921b1eddb88f7535e2f6318215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.255ex; height:2.843ex;" alt="{\displaystyle Y\mid \theta \sim P(Y\mid \theta )}"></span>. This parameter <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> is the SAT score of the student. The SAT score is viewed as a sample coming from a common population distribution indexed by another parameter <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, which is the high school grade of the student (freshman, sophomore, junior or senior).<sup id="cite_ref-FOOTNOTEGelmanCarlinSternRubin2004120–121_14-0" class="reference"><a href="#cite_note-FOOTNOTEGelmanCarlinSternRubin2004120–121-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> That 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 \theta \mid \phi \sim P(\theta \mid \phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo>∼</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \mid \phi \sim P(\theta \mid \phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669800b93e418b75d96dfe13173ab69af100242a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.479ex; height:2.843ex;" alt="{\displaystyle \theta \mid \phi \sim P(\theta \mid \phi )}"></span>. Moreover, the hyperparameter <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> follows its own distribution given by <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(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf70a07e977c536b7d1446ce59530c237ed35ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.94ex; height:2.843ex;" alt="{\displaystyle P(\phi )}"></span>, a hyperprior. To solve for the SAT score given information on the GPA, </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 ,\phi \mid Y)\propto P(Y\mid \theta ,\phi )P(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta ,\phi )P(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d0a57be0323f141e3802340a465ca47bad7708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.713ex; height:2.843ex;" alt="{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta ,\phi )P(\theta ,\phi )}"></span></dd> <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 ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02d1bed07edca33824a92c32821b930427fbde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.137ex; height:2.843ex;" alt="{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}"></span></dd></dl> <p>All information in the problem will be used to solve for the posterior distribution. Instead of solving only using the prior distribution and the likelihood function, the use of hyperpriors gives more information to make more accurate beliefs in the behavior of a parameter.<sup id="cite_ref-box_15-0" class="reference"><a href="#cite_note-box-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="2-stage_hierarchical_model">2-stage hierarchical model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=10" title="Edit section: 2-stage hierarchical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, the joint posterior distribution of interest in 2-stage hierarchical models is: </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 ,\phi \mid Y)={P(Y\mid \theta ,\phi )P(\theta ,\phi ) \over P(Y)}={P(Y\mid \theta )P(\theta \mid \phi )P(\phi ) \over P(Y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi \mid Y)={P(Y\mid \theta ,\phi )P(\theta ,\phi ) \over P(Y)}={P(Y\mid \theta )P(\theta \mid \phi )P(\phi ) \over P(Y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2485b195116224642dc670823167ace51a5834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.747ex; height:6.509ex;" alt="{\displaystyle P(\theta ,\phi \mid Y)={P(Y\mid \theta ,\phi )P(\theta ,\phi ) \over P(Y)}={P(Y\mid \theta )P(\theta \mid \phi )P(\phi ) \over P(Y)}}"></span></dd> <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 ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02d1bed07edca33824a92c32821b930427fbde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.137ex; height:2.843ex;" alt="{\displaystyle P(\theta ,\phi \mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi )}"></span><sup id="cite_ref-box_15-1" class="reference"><a href="#cite_note-box-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="3-stage_hierarchical_model">3-stage hierarchical model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=11" title="Edit section: 3-stage hierarchical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For 3-stage hierarchical models, the posterior distribution is given by: </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 ,\phi ,X\mid Y)={P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X) \over P(Y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>,</mo> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi ,X\mid Y)={P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X) \over P(Y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2628d63db93aed79a12e3c9c43cfe5053be6eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.439ex; height:6.509ex;" alt="{\displaystyle P(\theta ,\phi ,X\mid Y)={P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X) \over P(Y)}}"></span></dd> <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 ,\phi ,X\mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>,</mo> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∣</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta ,\phi ,X\mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1745b94ee2ea33ccf6bad0b622eb340ccf7b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.603ex; height:2.843ex;" alt="{\displaystyle P(\theta ,\phi ,X\mid Y)\propto P(Y\mid \theta )P(\theta \mid \phi )P(\phi \mid X)P(X)}"></span><sup id="cite_ref-box_15-2" class="reference"><a href="#cite_note-box-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Bayesian_nonlinear_mixed-effects_model">Bayesian nonlinear mixed-effects model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_hierarchical_modeling&action=edit&section=12" title="Edit section: Bayesian nonlinear mixed-effects model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bayesian_research_cycle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bayesian_research_cycle.png/500px-Bayesian_research_cycle.png" decoding="async" width="500" height="429" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bayesian_research_cycle.png/750px-Bayesian_research_cycle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bayesian_research_cycle.png/1000px-Bayesian_research_cycle.png 2x" data-file-width="2688" data-file-height="2304" /></a><figcaption>Bayesian research cycle using Bayesian nonlinear mixed effects model: (a) standard research cycle and (b) Bayesian-specific workflow <sup id="cite_ref-Lee2022_16-0" class="reference"><a href="#cite_note-Lee2022-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>.</figcaption></figure> <p>The framework of Bayesian hierarchical modeling is frequently used in diverse applications. Particularly, Bayesian nonlinear mixed-effects models have recently<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (November 2022)">when?</span></a></i>]</sup> received significant attention.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag may use weasel words or too-vague attribution. (November 2022)">by whom?</span></a></i>]</sup> A basic version of the Bayesian nonlinear mixed-effects models is represented as the following three-stage: </p><p><i><b>Stage 1: Individual-Level Model</b></i> </p><p><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}_{ij}=f(t_{ij};\theta _{1i},\theta _{2i},\ldots ,\theta _{li},\ldots ,\theta _{Ki})+\epsilon _{ij},\quad \epsilon _{ij}\sim N(0,\sigma ^{2}),\quad i=1,\ldots ,N,\,j=1,\ldots ,M_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {y}_{ij}=f(t_{ij};\theta _{1i},\theta _{2i},\ldots ,\theta _{li},\ldots ,\theta _{Ki})+\epsilon _{ij},\quad \epsilon _{ij}\sim N(0,\sigma ^{2}),\quad i=1,\ldots ,N,\,j=1,\ldots ,M_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a148cddfe5e8b961ad3c0bd572bda903c33bb03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:88.472ex; height:3.509ex;" alt="{\displaystyle {y}_{ij}=f(t_{ij};\theta _{1i},\theta _{2i},\ldots ,\theta _{li},\ldots ,\theta _{Ki})+\epsilon _{ij},\quad \epsilon _{ij}\sim N(0,\sigma ^{2}),\quad i=1,\ldots ,N,\,j=1,\ldots ,M_{i}.}"></span> </p><p><i><b>Stage 2: Population Model</b></i> </p><p><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 _{li}=\alpha _{l}+\sum _{b=1}^{P}\beta _{lb}x_{ib}+\eta _{li},\quad \eta _{li}\sim N(0,\omega _{l}^{2}),\quad i=1,\ldots ,N,\,l=1,\ldots ,K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{li}=\alpha _{l}+\sum _{b=1}^{P}\beta _{lb}x_{ib}+\eta _{li},\quad \eta _{li}\sim N(0,\omega _{l}^{2}),\quad i=1,\ldots ,N,\,l=1,\ldots ,K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc834ba3b45c354a74f4dc0dab3e070d226348d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:72.533ex; height:7.343ex;" alt="{\displaystyle \theta _{li}=\alpha _{l}+\sum _{b=1}^{P}\beta _{lb}x_{ib}+\eta _{li},\quad \eta _{li}\sim N(0,\omega _{l}^{2}),\quad i=1,\ldots ,N,\,l=1,\ldots ,K.}"></span> </p><p><i><b>Stage 3: Prior</b></i> </p><p><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 \sigma ^{2}\sim \pi (\sigma ^{2}),\quad \alpha _{l}\sim \pi (\alpha _{l}),\quad (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP})\sim \pi (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP}),\quad \omega _{l}^{2}\sim \pi (\omega _{l}^{2}),\quad l=1,\ldots ,K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <mi>π</mi> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>∼</mo> <mi>π</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>π</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>∼</mo> <mi>π</mi> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}\sim \pi (\sigma ^{2}),\quad \alpha _{l}\sim \pi (\alpha _{l}),\quad (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP})\sim \pi (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP}),\quad \omega _{l}^{2}\sim \pi (\omega _{l}^{2}),\quad l=1,\ldots ,K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b36041eea8accf23dbfd78c1dced7ceae3eb6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:105.695ex; height:3.343ex;" alt="{\displaystyle \sigma ^{2}\sim \pi (\sigma ^{2}),\quad \alpha _{l}\sim \pi (\alpha _{l}),\quad (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP})\sim \pi (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP}),\quad \omega _{l}^{2}\sim \pi (\omega _{l}^{2}),\quad l=1,\ldots ,K.}"></span> </p><p>Here, <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed6cfc23b1e9b298b8896b59780669e90b3f325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.616ex; height:2.343ex;" alt="{\displaystyle y_{ij}}"></span> denotes the continuous response of the <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th subject at the time point <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 t_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac4d83b980aeee694196ea954d449e2db972135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.317ex; height:2.676ex;" alt="{\displaystyle t_{ij}}"></span>, and <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_{ib}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{ib}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30d2b1b4d1c7261af28c383ab3542c420e8dc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.835ex; height:2.009ex;" alt="{\displaystyle x_{ib}}"></span> is the <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>-th covariate of the <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th subject. Parameters involved in the model are written in Greek letters. <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(t;\theta _{1},\ldots ,\theta _{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t;\theta _{1},\ldots ,\theta _{K})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc90a5afc9a8a37a858ba966d299f1b5cc7b27e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.068ex; height:2.843ex;" alt="{\displaystyle f(t;\theta _{1},\ldots ,\theta _{K})}"></span> is a known function parameterized by the <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>-dimensional vector <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 _{1},\ldots ,\theta _{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta _{1},\ldots ,\theta _{K})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6821b2c59c9d8e6b7d011cee289badbdc92e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.916ex; height:2.843ex;" alt="{\displaystyle (\theta _{1},\ldots ,\theta _{K})}"></span>. Typically, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is a `nonlinear' function and describes the temporal trajectory of individuals. In the model, <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 \epsilon _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e85834c4fe6c3f61e115a18433e23c93e6eb44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.421ex; height:2.343ex;" alt="{\displaystyle \epsilon _{ij}}"></span> and <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 \eta _{li}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{li}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725bbd45a3979050200eb94ff55b5844d0b7252f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.445ex; height:2.176ex;" alt="{\displaystyle \eta _{li}}"></span> describe within-individual variability and between-individual variability, respectively. If <i><b>Stage 3: Prior</b></i> is not considered, then the model reduces to a frequentist nonlinear mixed-effect model. </p><p><br /> A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate the posterior density: </p><p><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 \pi (\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K}|\{y_{ij}\}_{i=1,j=1}^{N,M_{i}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>,</mo> <mi>P</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K}|\{y_{ij}\}_{i=1,j=1}^{N,M_{i}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73b71337c2ae71d95fdd9d582502eb8de34ca77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:57.868ex; height:3.843ex;" alt="{\displaystyle \pi (\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K}|\{y_{ij}\}_{i=1,j=1}^{N,M_{i}})}"></span> </p><p><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 \propto \pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}},\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∝</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>,</mo> <mi>P</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \propto \pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}},\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3e7ee4d3e371f0c7954319050cbe8d785e8fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:60.709ex; height:3.843ex;" alt="{\displaystyle \propto \pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}},\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})}"></span> </p><p><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 =\underbrace {\pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}}|\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2})} _{Stage1:Individual-LevelModel}\times \underbrace {\pi (\{\theta _{li}\}_{i=1,l=1}^{N,K}|\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage2:PopulationModel}\times \underbrace {p(\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage3:Prior}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>t</mi> <mi>a</mi> <mi>g</mi> <mi>e</mi> <mn>1</mn> <mo>:</mo> <mi>I</mi> <mi>n</mi> <mi>d</mi> <mi>i</mi> <mi>v</mi> <mi>i</mi> <mi>d</mi> <mi>u</mi> <mi>a</mi> <mi>l</mi> <mo>−</mo> <mi>L</mi> <mi>e</mi> <mi>v</mi> <mi>e</mi> <mi>l</mi> <mi>M</mi> <mi>o</mi> <mi>d</mi> <mi>e</mi> <mi>l</mi> </mrow> </munder> <mo>×</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>K</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>,</mo> <mi>P</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>t</mi> <mi>a</mi> <mi>g</mi> <mi>e</mi> <mn>2</mn> <mo>:</mo> <mi>P</mi> <mi>o</mi> <mi>p</mi> <mi>u</mi> <mi>l</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mi>M</mi> <mi>o</mi> <mi>d</mi> <mi>e</mi> <mi>l</mi> </mrow> </munder> <mo>×</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>,</mo> <mi>P</mi> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>t</mi> <mi>a</mi> <mi>g</mi> <mi>e</mi> <mn>3</mn> <mo>:</mo> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>o</mi> <mi>r</mi> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\underbrace {\pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}}|\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2})} _{Stage1:Individual-LevelModel}\times \underbrace {\pi (\{\theta _{li}\}_{i=1,l=1}^{N,K}|\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage2:PopulationModel}\times \underbrace {p(\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage3:Prior}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5d532193dd3ea3158b602623e2e1a79ee62601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; margin-right: -0.028ex; width:113.732ex; height:7.676ex;" alt="{\displaystyle =\underbrace {\pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}}|\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2})} _{Stage1:Individual-LevelModel}\times \underbrace {\pi (\{\theta _{li}\}_{i=1,l=1}^{N,K}|\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage2:PopulationModel}\times \underbrace {p(\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage3:Prior}}"></span> </p><p><br /> The panel on the right displays Bayesian research cycle using Bayesian nonlinear mixed-effects model.<sup id="cite_ref-Lee2022_16-1" class="reference"><a href="#cite_note-Lee2022-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> A research cycle using the Bayesian nonlinear mixed-effects model comprises two steps: (a) standard research cycle and (b) Bayesian-specific workflow. Standard research cycle involves literature review, defining a problem and specifying the research question and hypothesis. Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle. </p> <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=Bayesian_hierarchical_modeling&action=edit&section=13" 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 reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-allenby-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-allenby_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-allenby_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Allenby, Rossi, McCulloch (January 2005). <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=655541">"Hierarchical Bayes Model: A Practitioner’s Guide"</a>. <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/228167480_Bayesian_Applications_in_Marketing">Journal of Bayesian Applications in Marketing</a>, pp. 1–4. Retrieved 26 April 2014, p. 3</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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="CITEREFGelmanCarlinSternRubin2004" class="citation book cs1"><a href="/wiki/Andrew_Gelman" title="Andrew Gelman">Gelman, Andrew</a>; Carlin, John B.; Stern, Hal S. & Rubin, Donald B. 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Boca Raton, Florida: CRC Press. pp. 4–5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-388-X" title="Special:BookSources/1-58488-388-X"><bdi>1-58488-388-X</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+Data+Analysis&rft.place=Boca+Raton%2C+Florida&rft.pages=4-5&rft.edition=second&rft.pub=CRC+Press&rft.date=2004&rft.isbn=1-58488-388-X&rft.aulast=Gelman&rft.aufirst=Andrew&rft.au=Carlin%2C+John+B.&rft.au=Stern%2C+Hal+S.&rft.au=Rubin%2C+Donald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+hierarchical+modeling" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeeLeiMallick2020" class="citation journal cs1">Lee, Se Yoon; Lei, Bowen; Mallick, Bani (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7390340">"Estimation of COVID-19 spread curves integrating global data and borrowing information"</a>. <i>PLOS ONE</i>. <b>15</b> (7): e0236860. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2005.00662">2005.00662</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0236860">10.1371/journal.pone.0236860</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7390340">7390340</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32726361">32726361</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Estimation+of+COVID-19+spread+curves+integrating+global+data+and+borrowing+information&rft.volume=15&rft.issue=7&rft.pages=e0236860&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7390340%23id-name%3DPMC&rft_id=info%3Apmid%2F32726361&rft_id=info%3Aarxiv%2F2005.00662&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0236860&rft.aulast=Lee&rft.aufirst=Se+Yoon&rft.au=Lei%2C+Bowen&rft.au=Mallick%2C+Bani&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7390340&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+hierarchical+modeling" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeeMallick2021" class="citation journal cs1">Lee, Se Yoon; Mallick, Bani (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13571-020-00245-8">"Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas"</a>. <i>Sankhya B</i>. <b>84</b>: 1–43. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13571-020-00245-8">10.1007/s13571-020-00245-8</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sankhya+B&rft.atitle=Bayesian+Hierarchical+Modeling%3A+Application+Towards+Production+Results+in+the+Eagle+Ford+Shale+of+South+Texas&rft.volume=84&rft.pages=1-43&rft.date=2021&rft_id=info%3Adoi%2F10.1007%2Fs13571-020-00245-8&rft.aulast=Lee&rft.aufirst=Se+Yoon&rft.au=Mallick%2C+Bani&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs13571-020-00245-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+hierarchical+modeling" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGelmanCarlinSternRubin20046-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin20046_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGelmanCarlinSternRubin2004">Gelman et al. 2004</a>, p. 6.</span> </li> <li id="cite_note-FOOTNOTEGelmanCarlinSternRubin2004117-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin2004117_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin2004117_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGelmanCarlinSternRubin2004">Gelman et al. 2004</a>, p. 117.</span> </li> <li id="cite_note-good-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-good_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-good_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGood1980" class="citation journal cs1">Good, I.J. (1980). <a rel="nofollow" class="external text" href="http://dialnet.unirioja.es/servlet/oaiart?codigo=2368428">"Some history of the hierarchical Bayesian methodology"</a>. <i>Trabajos de Estadistica y de Investigacion Operativa</i>. <b>31</b>: 489–519. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02888365">10.1007/BF02888365</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:121270218">121270218</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trabajos+de+Estadistica+y+de+Investigacion+Operativa&rft.atitle=Some+history+of+the+hierarchical+Bayesian+methodology&rft.volume=31&rft.pages=489-519&rft.date=1980&rft_id=info%3Adoi%2F10.1007%2FBF02888365&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121270218%23id-name%3DS2CID&rft.aulast=Good&rft.aufirst=I.J.&rft_id=http%3A%2F%2Fdialnet.unirioja.es%2Fservlet%2Foaiart%3Fcodigo%3D2368428&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+hierarchical+modeling" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Bernardo, Smith(1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=11nSgIcd7xQC&dq=bernardo+degroot+lindley&pg=PA497">Bayesian Theory</a>. Chichester, England: John Wiley & Sons, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-92416-4" title="Special:BookSources/0-471-92416-4">0-471-92416-4</a>, p. 23</span> </li> <li id="cite_note-FOOTNOTEGelmanCarlinSternRubin20046–8-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin20046–8_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin20046–8_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGelmanCarlinSternRubin2004">Gelman et al. 2004</a>, pp. 6–8.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Bernardo, Degroot, Lindley (September 1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=myfRtgAACAAJ&q=Proceedings+of+the+Second+Valencia+International+Meeting">“Proceedings of the Second Valencia International Meeting”</a>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting">Bayesian Statistics 2</a>. 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Annals of Probability, pp. 745–747</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Bernardo, Degroot, Lindley (September 1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting">“Proceedings of the Second Valencia International Meeting”</a>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting">Bayesian Statistics 2</a>. Amsterdam: Elsevier Science Publishers B.V, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-87746-0" title="Special:BookSources/0-444-87746-0">0-444-87746-0</a>, pp. 371–372</span> </li> <li id="cite_note-FOOTNOTEGelmanCarlinSternRubin2004120–121-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGelmanCarlinSternRubin2004120–121_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGelmanCarlinSternRubin2004">Gelman et al. 2004</a>, pp. 120–121.</span> </li> <li id="cite_note-box-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-box_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-box_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-box_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/George_E._P._Box" title="George E. P. Box">Box G. E. P.</a>, <a href="/w/index.php?title=George_Tiao&action=edit&redlink=1" class="new" title="George Tiao (page does not exist)">Tiao G. C.</a> (1965). <a rel="nofollow" class="external text" href="http://projecteuclid.org/download/pdf_1/euclid.aoms/1177699906">"Multiparameter problem from a bayesian point of view"</a>. <a rel="nofollow" class="external text" href="http://projecteuclid.org/all/euclid.aoms">Multiparameter Problems From A Bayesian Point of View Volume 36 Number 5</a>. New York City: John Wiley & Sons, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-57428-7" title="Special:BookSources/0-471-57428-7">0-471-57428-7</a></span> </li> <li id="cite_note-Lee2022-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lee2022_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lee2022_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2022" class="citation journal cs1">Lee, Se Yoon (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath10060898">"Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications"</a>. <i>Mathematics</i>. <b>10</b> (6): 898. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2201.12430">2201.12430</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath10060898">10.3390/math10060898</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics&rft.atitle=Bayesian+Nonlinear+Models+for+Repeated+Measurement+Data%3A+An+Overview%2C+Implementation%2C+and+Applications&rft.volume=10&rft.issue=6&rft.pages=898&rft.date=2022&rft_id=info%3Aarxiv%2F2201.12430&rft_id=info%3Adoi%2F10.3390%2Fmath10060898&rft.aulast=Lee&rft.aufirst=Se+Yoon&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fmath10060898&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+hierarchical+modeling" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Bayesian_hierarchical_modeling&oldid=1249523057">https://en.wikipedia.org/w/index.php?title=Bayesian_hierarchical_modeling&oldid=1249523057</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_networks" title="Category:Bayesian networks">Bayesian networks</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:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_vague_or_ambiguous_time" title="Category:All articles with vague or ambiguous time">All articles with vague or ambiguous time</a></li><li><a href="/wiki/Category:Vague_or_ambiguous_time_from_November_2022" title="Category:Vague or ambiguous time from November 2022">Vague or ambiguous time from November 2022</a></li><li><a href="/wiki/Category:Articles_with_specifically_marked_weasel-worded_phrases_from_November_2022" title="Category:Articles with specifically marked weasel-worded phrases from November 2022">Articles with specifically marked weasel-worded phrases from November 2022</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 5 October 2024, at 10:49<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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Bayesian hierarchical modeling - Wikipedia