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February 2017 Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with $p>N$
Tavis Abrahamsen, James P. Hobert
Bernoulli 23(1): 459-478 (February 2017). DOI: 10.3150/15-BEJ749


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. Román and Hobert [Linear Algebra Appl. 473 (2015) 54–77] 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, Román and Hobert’s (2015) result is extended by allowing improper priors on the variance components, and, more importantly, by removing all assumptions on the $X$ matrix. So, not only is $X$ 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 Román and Hobert’s (2015) proof. Consequently, the extension to unrestricted $X$ requires a substantially different analysis.


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Tavis Abrahamsen. James P. Hobert. "Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with $p>N$." Bernoulli 23 (1) 459 - 478, February 2017.


Received: 1 February 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1368.62061
MathSciNet: MR3556779
Digital Object Identifier: 10.3150/15-BEJ749

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability


Vol.23 • No. 1 • February 2017
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