Block Gibbs samplers for logistic mixed models: convergence properties and a comparison with full Gibbs samplers

Logistic linear mixed model (LLMM) is one of the most widely used statistical models. Generally, Markov chain Monte Carlo algorithms are used to explore the posterior densities associated with Bayesian LLMMs. Polson, Scott and Windle's (2013) Polya-Gamma data augmentation (DA) technique can be used to construct full Gibbs (FG) samplers for LLMMs. Here, we develop efficient block Gibbs (BG) samplers for Bayesian LLMMs using the P\'{o}lya-Gamma DA method. We compare the FG and BG samplers in the context of simulated and real data examples as the correlation between the fixed and random effects change as well as when the dimension of the design matrices varies. These numerical examples demonstrate superior performance of the BG samplers over the FG samplers. We also consider conditions guaranteeing geometric ergodicity of the BG Markov chain when an improper uniform prior is assigned on the regression coefficients and proper or improper priors are placed on the variance parameters of the random effects. This theoretical result has important practical implications, including honest statistical inference with valid Monte Carlo standard errors.