Dynamic causal modeling: Difference between revisions

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== Bayesian model reduction ==

Bayesian model reduction <ref name=":0">{{Cite journal|last=Friston|first=Karl|last2=Penny|first2=Will|date=June 2011|title=Post hoc Bayesian model selection|url=https://doi.org/10.1016/j.neuroimage.2011.03.062|journal=NeuroImage|volume=56|issue=4|pages=2089–2099|doi=10.1016/j.neuroimage.2011.03.062|issn=1053-8119|pmc=PMC3112494|pmid=21459150|via=}}</ref><ref name=":1">{{Cite journal|last=Friston|first=Karl J.|last2=Litvak|first2=Vladimir|last3=Oswal|first3=Ashwini|last4=Razi|first4=Adeel|last5=Stephan|first5=Klaas E.|last6=van Wijk|first6=Bernadette C.M.|last7=Ziegler|first7=Gabriel|last8=Zeidman|first8=Peter|date=March 2016|title=Bayesian model reduction and empirical Bayes for group (DCM) studies|url=https://doi.org/10.1016/j.neuroimage.2015.11.015|journal=NeuroImage|volume=128|pages=413–431|doi=10.1016/j.neuroimage.2015.11.015|issn=1053-8119|pmc=PMC4767224|pmid=26569570|via=}}</ref> is a method for computing the [[Marginal likelihood|evidence]] and [[Posterior probability|posterior]] over the parameters of [[Bayesian statistics|Bayesian]] models whichthat differ in the specification of their [[Prior probability|priors]]. A full model is fitted to the available data using standard approaches. Hypotheses are then tested by defining one or more 'reduced' models with alternative (and usually more restrictive) priors, which usually – in the limit will– switch off certain parameters. The evidence and parameters of the reduced models can then be computed from the evidence and estimated ([[Posterior probability|posterior]]) parameters of the full model using Bayesian model reduction. If the priors and posteriors are [[Normal distribution|normally distributed]], then there is an analytic solution which can be computed rapidly. This has multiple scientific and engineering applications,: includingthese rapidlyinclude scoring the evidence for large numbers of models very quickly and facilitating the estimation of hierarchical models ([[Empirical Bayes method|Parametric Empirical Bayes]]).

== Theory ==

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Where the tilde symbol (~) indicates quantities relating to the reduced model and subscript zero - such as <math>\mu_{0}</math> - indicates parameters of the priors. For convenience we also define precision matrices, which are simply the inverse of each covariance matrix:

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We also assume that theThe free energy of the full model <math>F</math> has been computed, which is an approximation (lower bound) ofon the log model evidence: <math>F\approx \ln{p(y)}</math> that is optimised explicitly in variational Bayes (or can be recovered from sampling approximations). The reduced model's free energy <math>\tilde{F}</math> and parameters <math>(\tilde{\mu},\tilde{\Sigma})</math> are then given by the expressions:

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