Analysis of the Polya-Gamma block Gibbs sampler for Bayesian logistic linear mixed models

TL;DR

This paper develops and analyzes a two-block Gibbs sampler using Polya-Gamma data augmentation for Bayesian logistic linear mixed models, proving its uniform ergodicity and ensuring reliable MCMC estimates.

Contribution

It introduces a new Gibbs sampling algorithm for Bayesian logistic mixed models and proves its uniform ergodicity, enhancing theoretical understanding and practical reliability.

Findings

The Gibbs sampler is uniformly ergodic.

Central limit theorems hold for MCMC estimators.

The method improves convergence guarantees for Bayesian logistic models.

Abstract

In this article, we construct a two-block Gibbs sampler using Polson et al. (2013) data augmentation technique with Polya-Gamma latent variables for Bayesian logistic linear mixed models under proper priors. Furthermore, we prove the uniform ergodicity of this Gibbs sampler, which guarantees the existence of the central limit theorems for MCMC based estimators.