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Exploration of the intractable posterior distributions associated withBayesian versions of the general linear mixed model is often performed usingMarkov chain Monte Carlo. In particular, if a conditionally conjugate prior isused, then there is a simple two-block Gibbs sampler available. Rom\'{a}n andHobert [Linear Algebra Appl. 473 (2015) 54-77] showed that, when the priors areproper and the $X$ matrix has full column rank, the Markov chains underlyingthese Gibbs samplers are nearly always geometrically ergodic. In this paper,Rom\'{a}n and Hobert's (2015) result is extended by allowing improper priors onthe variance components, and, more importantly, by removing all assumptions onthe $X$ matrix. So, not only is $X$ allowed to be (column) rank deficient,which provides additional flexibility in parameterizing the fixed effects, itis also allowed to have more columns than rows, which is necessary in theincreasingly important situation where $p>N$. The full rank assumption on $X$is at the heart of Rom\'{a}n and Hobert's (2015) proof. Consequently, theextension to unrestricted $X$ requires a substantially different analysis.

Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with $p>N$