Dimension free convergence rates for Gibbs samplers for Bayesian linear mixed models

TL;DR

This paper investigates how the convergence rates of Gibbs samplers for Bayesian linear mixed models behave as data size increases, revealing dimension-free convergence properties using Wasserstein techniques.

Contribution

It extends previous convergence analysis from simple models to more complex Bayesian mixed models, demonstrating dimension-free convergence rates.

Findings

Gibbs samplers for Bayesian mixed models exhibit dimension-free convergence rates.

Wasserstein-based techniques effectively analyze convergence behavior.

Convergence rate approaches zero as data size grows, similar to simpler models.

Abstract

The emergence of big data has led to a growing interest in so-called convergence complexity analysis, which is the study of how the convergence rate of a Monte Carlo Markov chain (for an intractable Bayesian posterior distribution) scales as the underlying data set grows in size. Convergence complexity analysis of practical Monte Carlo Markov chains on continuous state spaces is quite challenging, and there have been very few successful analyses of such chains. One fruitful analysis was recently presented by Qin and Hobert (2021b), who studied a Gibbs sampler for a simple Bayesian random effects model. These authors showed that, under regularity conditions, the geometric convergence rate of this Gibbs sampler converges to zero as the data set grows in size. It is shown herein that similar behavior is exhibited by Gibbs samplers for more general Bayesian models that possess both random effects and traditional continuous covariates, the so-called mixed models. The analysis employs the Wasserstein-based techniques introduced by Qin and Hobert (2021b).