Post-hoc selection of dynamic causal models

M.J Rosa,K Friston,W Penny

doi:10.1016/j.jneumeth.2012.04.013

Abstract

Dynamic causal modelling (DCM) was originally proposed as a hypothesis driven procedure in which a small number of neurobiologically motivated models are compared. Model comparison in this context usually proceeds by individually fitting each model to data and then approximating the corresponding model evidence with a free energy bound. However, a recent trend has emerged for comparing very large numbers of models in a more exploratory manner. This led Friston and Penny (2011) to propose a post-hoc approximation to the model evidence, which is computed by optimising only the largest (full) model of a set of models. The evidence for any (reduced) submodel is then obtained using a generalisation of the Savage-Dickey density ratio (Dickey, 1971). The benefit of this post-hoc approach is a huge reduction in the computational time required for model fitting. This is because only a single model is fitted to data, allowing a potentially huge model space to be searched relatively quickly. In this paper, we explore the relationship between the free energy bound and post-hoc approximations to the model evidence in the context of deterministic (bilinear) dynamic causal models (DCMs) for functional magnetic resonance imaging data.

Highlights

Bayesian model selection (BMS) is a powerful method to compare different models for explaining observed data
Dynamic causal modelling is a mathematical framework to estimate, and make inferences about, the coupling among brain areas and how this coupling is influenced by changes in experimental context (Friston et al, 2003)
Our results show very good agreement between optimised and post-hoc model evidences, for both synthetic and real functional magnetic resonance imaging (fMRI) data

Summary

Introduction

Bayesian model selection (BMS) is a powerful method to compare different models for explaining observed data. BMS is based on the model evidence, which is the probability of obtaining a particular model, given the data Even though this quantity is not, in general, straightforward to compute, it is well established that statistical models can be compared using a variational free energy approximation to the evidence (Beal and Ghahramani, 2003). This approximation has widespread application, and, in neuroimaging, it has become the method of choice for comparing models of effective brain connectivity, in particular dynamic causal models (DCMs) (Stephan et al, 2010; Penny, 2012). It was originally introduced as a hypothesis driven procedure, in which a small number of neurobiologically motivated models are compared, recently, a trend has emerged for comparing very large numbers of models in a more exploratory manner

Methods

Results

Conclusion