Efficiently Generating Independent Samples Directly from the Posterior Distribution for a Large Class of Bayesian Generalized Linear Mixed Effects Models

Markov chain Monte Carlo (MCMC) is an all-purpose tool that allows one to generate dependent replicates from a posterior distribution for effectively any Bayesian hierarchical model. As such, MCMC has become a standard in Bayesian statistics. However, convergence issues, tuning, and the effective sample size of the MCMC are nontrivial considerations that are often overlooked or can be difficult to assess. Moreover, these practical issues can produce a significant computational burden. This motivates us to consider finding closed-form expressions of the posterior distribution that are computationally straightforward to sample from directly. We focus on a broad class of Bayesian generalized linear mixed-effects models (GLMM) that allows one to jointly model data of different types (e.g., Gaussian, Poisson, and binomial distributed observations). Exact sampling from the posterior distribution for Bayesian GLMMs is such a difficult problem that it is now arguably overlooked as a possible problem to solve. To solve this problem, we derive a new class of distributions that gives one the flexibility to specify the prior on fixed and random effects to be any conjugate multivariate distribution. We refer to this new distribution as the generalized conjugate multivariate (GCM) distribution, and several technical results are provided. The expression of the exact posterior distribution is given along with the steps to obtain direct independent simulations from the posterior distribution. These direct simulations have an efficient projection/regression form, and hence, we refer to our method as Exact Posterior Regression (EPR). Several theoretical results are developed that create the foundation for EPR. Illustrative examples are provided including a simulation study and an analysis of estimates from the U.S. Census Bureau's American Community Survey (ACS).