Dynamic causal modeling: Difference between revisions

Revision as of 11:20, 10 May 2018

Revision as of 11:49, 10 May 2018

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==== Gaussian priors and posteriors ====

Under Gaussian prior and posterior densities, as are often used in the context of variational Bayes to approximate a true posterior with non-standard form, Bayesian model reduction has ana simple analytical expression. DefineWe define normal densities for the priors and posteriors:

<math>\begin{align}

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\end{align}</math>

Where the tilde symbol (~) indicates quantities relating to the reduced model. 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:

<math>\begin{align}

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\Pi_0&=\Sigma_0^{-1}\\

\tilde{\Pi}&=\tilde{\Sigma}^{-1}\\

\tilde{\Pi}_0&=\tilde{\Sigma_0Sigma}_0^{-1}\\

\end{align}</math>

TheWe also assume that the free energy of the full model has been computed, which is a lower bound on the log model evidence: <math>F\approx \ln{p(y)}</math>. The reduced model's free energy <math>\tilde{F}</math> and parameters <math>(\tilde{\mu},\tilde{\Sigma})</math> isare then given by the expression:

<math>\begin{align}

\tilde{F } &= \frac{1}{2}\ln|\tilde{\Pi}_0\cdot\Pi\cdot\tilde{\Sigma}\cdot\Sigma_0| \\

&- \frac{1}{2}(\mu^T\Pi\mu + \tilde{\mu}_0^T\tilde{\Pi}_0\tilde{\mu}_0 - \mu_0^T\Pi_0\mu_0 - \tilde{\mu}^T\tilde{\Sigma}\tilde{\mu}) + F\\

\tilde{\mu} &= \tilde{\Sigma}(\Pi\mu + \tilde{\Pi}_0\tilde{\mu}_0+\Pi_0\mu_0) \\

\tilde{\Sigma} &= (\Pi+\tilde{\Pi}_0-\Pi_0)^{-1} \\

\end{align}</math>