Asymptotic normality of posterior distributions for generalized linear mixed models

Bayesian inference methods are used extensively in the analysis of Generalized Linear Mixed Models (GLMMs), but it may be difficult to handle the posterior distributions analytically. In this paper, we establish the asymptotic normality of the joint posterior distribution of the parameters and the random effects in a GLMM by using Stein’s Identity. We also show that while incorrect assumptions on the random effects can lead to substantial bias in the estimates of the parameters, the assumed model for the random effects, under some regularity conditions, does not affect the asymptotic normality of the joint posterior distribution. This motivates the use of the approximate normal distributions for sensitivity analysis of the random effects distribution. We additionally illustrate that the approximate normal distribution performs reasonably using both real and simulated data. This creates a primary alternative to Markov Chain Monte Carlo (MCMC) sampling and avoids a wide range of problems for MCMC algorithms in terms of convergence and computational time.