Bayesian model reduction

TL;DR

This paper reviews Bayesian model reduction, a technique for efficiently computing evidence and parameters of probabilistic models with different priors, with applications in neuroimaging and neuroscience.

Contribution

It provides a comprehensive review, equations, and worked examples of Bayesian model reduction for various distributions and models, including neurobiological applications.

Findings

Analytical solutions for Bayesian model reduction in variational Bayes.

Worked examples in multivariate regression, Gaussian mixtures, and dynamical systems.

Applications demonstrated in neuroimaging and neuroscience contexts.

Abstract

This paper reviews recent developments in statistical structure learning; namely, Bayesian model reduction. Bayesian model reduction is a method for rapidly computing the evidence and parameters of probabilistic models that differ only in their priors. In the setting of variational Bayes this has an analytical solution, which finesses the problem of scoring large model spaces in model comparison or structure learning. In this technical note, we review Bayesian model reduction and provide the relevant equations for several discrete and continuous probability distributions. We provide worked examples in the context of multivariate linear regression, Gaussian mixture models and dynamical systems (dynamic causal modelling). These examples are accompanied by the Matlab scripts necessary to reproduce the results. Finally, we briefly review recent applications in the fields of neuroimaging and neuroscience. Specifically, we consider structure learning and hierarchical or empirical Bayes that can be regarded as a metaphor for neurobiological processes like abductive reasoning.