Model Selection in Variational Mixed Effects Models

Abstract: 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.