  This note is concerned with an accurate and computationally efficient
variational bayesian treatment of mixed-effects modelling. We focus on group
studies, i.e. empirical studies that report multiple measurements acquired in
multiple subjects. When approached from a bayesian perspective, such
mixed-effects models typically rely upon a hierarchical generative model of the
data, whereby both within- and between-subject effects contribute to the
overall observed variance. The ensuing VB scheme can be used to assess
statistical significance at the group level and/or to capture inter-individual
differences. Alternatively, it can be seen as an adaptive regularization
procedure, which iteratively learns the corresponding within-subject priors
from estimates of the group distribution of effects of interest (cf. so-called
"empirical bayes" approaches). We outline the mathematical derivation of the
ensuing VB scheme, whose open-source implementation is available as part the
VBA toolbox.
