Dynamic causal modeling

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Bayesian model reduction

Bayesian model reduction is a method for computing the evidence and parameters of Bayesian models which differ only in the specification of their priors. Typically, a 'full' model is fitted to the available data with standard approaches. Then, hypotheses are tested by defining one or more 'reduced' models, which differ only in their priors. In the context of variational Bayes, the evidence and parameters of the reduced models can be computed analytically from the evidence and parameters of the full model. This has numerous applications, including rapidly scoring large numbers of models, and facilitating the estimation of hierarchical (Parametric Empirical Bayes) models.

Theory

Consider some model with parameters ${\displaystyle \theta }$  and a prior probability density on those parameters ${\displaystyle P(\theta )}$ . The posterior belief about ${\displaystyle \theta }$  after seeing the data ${\displaystyle p(\theta |y)}$  is given by Bayes rule:

${\displaystyle {\begin{aligned}p(\theta |y)&={\frac {p(y|\theta )p(\theta )}{p(y)}}\\p(y)&=\int p(y|\theta )p(\theta )d\theta \end{aligned}}}$ 

The second line defines the model evidence, which is the probability of observing the given data under a model with these parameters. In practice, the posterior cannot usually be computed analytically due to the difficult integral. Therefore, the posteriors are estimated using approaches such as MCMC sampling or variational Bayes. Having estimated the posteriors and evidence using one of these approaches, we can next define a reduced model with an alternative set of priors