  The generalized linear mixed model (GLMM) is a popular statistical approach
for handling correlated data, and is used extensively in applications areas
where big data is common, including biomedical data settings. The focus of this
paper is scalable statistical inference for the GLMM, where we define
statistical inference as: (i) estimation of population parameters, and (ii)
evaluation of scientific hypotheses in the presence of uncertainty. Artificial
intelligence (AI) learning algorithms excel at scalable statistical estimation,
but rarely include uncertainty quantification. In contrast, Bayesian inference
provides full statistical inference, since uncertainty quantification results
automatically from the posterior distribution. Unfortunately, Bayesian
inference algorithms, including Markov Chain Monte Carlo (MCMC), become
computationally intractable in big data settings. In this paper, we introduce a
statistical inference algorithm at the intersection of AI and Bayesian
inference, that leverages the scalability of modern AI algorithms with
guaranteed uncertainty quantification that accompanies Bayesian inference. Our
algorithm is an extension of stochastic gradient MCMC with novel contributions
that address the treatment of correlated data (i.e., intractable marginal
likelihood) and proper posterior variance estimation. Through theoretical and
empirical results we establish our algorithm's statistical inference
properties, and apply the method in a large electronic health records database.
