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

TL;DR

This paper introduces efficient block Gibbs samplers for Bayesian logistic linear mixed models using Polya-Gamma data augmentation, demonstrating their superior performance over full Gibbs samplers and establishing conditions for their geometric ergodicity.

Contribution

The paper develops and compares block Gibbs samplers with full Gibbs samplers for Bayesian LLMMs, providing theoretical guarantees for their convergence.

Findings

Block Gibbs samplers outperform full Gibbs samplers in numerical experiments.

Geometric ergodicity of the block Gibbs chain is established under common priors.

Theoretical justification for using Monte Carlo standard errors in posterior estimation.

Abstract

The 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 the Bayesian LLMMs. Polson, Scott and Windle's (2013) Polya-Gamma data augmentation (DA) technique can be used to construct full Gibbs (FG) samplers for the LLMMs. Here, we develop efficient block Gibbs (BG) samplers for Bayesian LLMMs using the Polya-Gamma DA method. We compare the FG and BG samplers in the context of a real data example, as the correlation between the fixed effects and the random effects changes as well as when the dimensions of the design matrices vary. These numerical examples demonstrate superior performance of the BG samplers over the FG samplers. We also derive conditions guaranteeing geometric ergodicity of the BG Markov chain when the popular 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 as it justifies the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates of the posterior quantities.