Convergence Analysis of Block Gibbs Samplers for Bayesian Linear Mixed Models with p > N

Exploration of the intractable posterior distributions associated with Bayesian versions of the general linear mixed model is often performed using Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is used, then there is a simple two-block Gibbs sampler available. Roman & Hobert (2015) showed that, when the priors are proper and the X matrix has full column rank, the Markov chains underlying these Gibbs samplers are nearly always geometrically ergodic. In this paper, Roman & Hobert’s (2015) result is extended by allowing improper priors on the variance components, and, more importantly, by removing all assumptions on theX matrix. So, not only isX allowed to be (column) rank deficient, which provides additional flexibility in parameterizing the fixed effects, it is also allowed to have more columns than rows, which is necessary in the increasingly important situation where p > N . The full rank assumption on X is at the heart of Roman & Hobert’s (2015) proof. Consequently, the extension to unrestrictedX requires a substantially different analysis.