Abstract:
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The general linear mixed model (GLMM) is an extremely flexible class of models. When using proper, conditionally conjugate priors, the Bayesian version of this model admits a simple two-block Gibbs sampler that can be used to explore the corresponding intractable posterior density. Roman & Hobert (2015) show that under mild conditions the block Gibbs Markov chain is geometrically ergodic. However, their results assume that the covariate matrix X is full rank. We extend their results to the case where X is completely unrestricted. That is, not only do we allow X to be rank deficient, but we also allow for the case where there are more predictors than data points, p > N. The full rank assumption on X is at the very heart of Roman & Hobert's proof, and thus our analysis is significantly different than theirs. Surprisingly, we are able to recover the same sufficient conditions for geometric ergodicity as them even though we use an entirely different drift function for our analysis. Lastly, we extend our results even further by providing sufficient conditions for the block Gibbs sampler to be geometrically ergodic under a class of partially proper priors.
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