Scalable Bayesian Inference for Generalized Linear Mixed Models via Stochastic Gradient MCMC

The generalized linear mixed model (GLMM) is widely used for analyzing correlated data, particularly in large-scale biomedical and social science applications. Scalable Bayesian inference for GLMMs is challenging because the marginal likelihood is intractable and conventional Markov chain Monte Carlo (MCMC) methods become computationally prohibitive as the number of subjects grows. We develop a stochastic gradient MCMC (SGMCMC) algorithm tailored to GLMMs that enables accurate posterior inference in the large-sample regime. Our approach uses Fisher's identity to construct an unbiased Monte Carlo estimator of the gradient of the marginal log-likelihood, making SGMCMC feasible when direct gradient computation is impossible. We analyze the additional variability introduced by both minibatching and gradient approximation, and derive a post-hoc covariance correction that yields properly calibrated posterior uncertainty. Through simulations, we show that the proposed method provides accurate posterior means and variances, outperforming existing approaches, including control variate methods, in large-$n$ settings. We further demonstrate the method's practical utility in an analysis of electronic health records data, where accounting for variance inflation materially changes scientific conclusions.