Structured Spike-and-Slab Variational Bayes for High-Dimensional Non-Normal Generalized Linear Mixed Models

Abstract

High-dimensional non-normal longitudinal data are ubiquitous across fields such as genomics, biomedicine, microbiome research, and the social sciences. Such data often combine non-normal responses, within-subject dependence and sparse population effects. We develop a structured variational Bayesian procedure: variational Bayesian empirical likelihood with spike-and-slab priors, which combines an empirical-likelihood-motivated working kernel with two Laplace shrinkage branches. The variational family links each inclusion indicator to its coefficient and local scale, and it uses probability-weighted updates for global shrinkage. A scalar empirical-likelihood weighting implementation provides tractable computation, and conditional working-evidence scores compare random-effect covariance structures. Simulation studies examine selection, estimation, prediction, interval coverage, and sensitivity to numerical and prior settings. Two longitudinal microbiome applications yield interpretable conditional associations. The resulting framework provides an explicit computational construction for sparse mixed-model analysis, with numerical results characterizing its operating behavior.