  Variational inference is an alternative estimation technique for Bayesian
models. Recent work shows that variational methods provide consistent
estimation via efficient, deterministic algorithms. Other tools, such as model
selection using variational AICs (VAIC) have been developed and studied for the
linear regression case. While mixed effects models have enjoyed some study in
the variational context, tools for model selection are lacking. One important
feature of model selection in mixed effects models, particularly longitudinal
models, is the selection of the random effects which in turn determine the
covariance structure for the repeatedly sampled outcome. To address this, we
derive a VAIC specifically for variational mixed effects (VME) models. We also
implement a parameter-efficient VME as part of our study which reduces any
general random effects structure down to a single subject-specific score. This
model accommodates a wide range of random effect structures including random
intercept and slope models as well as random functional effects. Our VAIC can
model and perform selection on a variety of VME models including more classic
longitudinal models as well as longitudinal scalar-on-function regression. As
we demonstrate empirically, our VAIC performs well in discriminating between
correctly and incorrectly specified random effects structures. Finally, we
illustrate the use of VAICs for VMEs on two datasets: a study of lead levels in
children and a study of diffusion tensor imaging.
