Empirical Bayes for DCM: A Group Inversion Scheme

This technical note considers a simple but important methodological issue in estimating effective connectivity; namely, how do we integrate measurements from multiple subjects to infer functional brain architectures that are conserved over subjects. We offer a solution to this problem that rests on a generalisation of random effects analyses to Bayesian inference about nonlinear models of electrophysiological time-series data. Specifically, we present an empirical Bayesian scheme for group or hierarchical models, in the setting of dynamic causal modelling (DCM). Recent developments in approximate Bayesian inference for hierarchical models enable the efficient estimation of group effects in DCM studies of multiple trials, sessions or subjects. This approach estimates second (e.g., between-subject) level parameters based posterior estimates from the first (e.g., within-subject) level. Here, we use empirical priors from the second level to iteratively optimise posterior densities over parameters at the first level. The motivation for this iterative application is to finesse the local minima problem inherent in the (first level) inversion of nonlinear and ill-posed models. Effectively, the empirical priors shrink the first level parameter estimates towards the global maximum, to provide more robust and efficient estimates of within (and between-subject) effects. This paper describes the inversion scheme using a worked example based upon simulated electrophysiological responses. In the subsequent paper, we will assess its robustness and reproducibility using an empirical example.