  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 (indicating immediate
convergence) 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).
