Open Access
December 2012 Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors
Jorge Carlos Román, James P. Hobert
Ann. Statist. 40(6): 2823-2849 (December 2012). DOI: 10.1214/12-AOS1052


Bayesian analysis of data from the general linear mixed model is challenging because any nontrivial prior leads to an intractable posterior density. However, if a conditionally conjugate prior density is adopted, then there is a simple Gibbs sampler that can be employed to explore the posterior density. A popular default among the conditionally conjugate priors is an improper prior that takes a product form with a flat prior on the regression parameter, and so-called power priors on each of the variance components. In this paper, a convergence rate analysis of the corresponding Gibbs sampler is undertaken. The main result is a simple, easily-checked sufficient condition for geometric ergodicity of the Gibbs–Markov chain. This result is close to the best possible result in the sense that the sufficient condition is only slightly stronger than what is required to ensure posterior propriety. The theory developed in this paper is extremely important from a practical standpoint because it guarantees the existence of central limit theorems that allow for the computation of valid asymptotic standard errors for the estimates computed using the Gibbs sampler.


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Jorge Carlos Román. James P. Hobert. "Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors." Ann. Statist. 40 (6) 2823 - 2849, December 2012.


Published: December 2012
First available in Project Euclid: 8 February 2013

zbMATH: 1296.60204
MathSciNet: MR3097961
Digital Object Identifier: 10.1214/12-AOS1052

Primary: 60J27
Secondary: 62F15

Keywords: convergence rate , geometric drift condition , geometric ergodicity , Markov chain , Monte Carlo , posterior propriety

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • December 2012
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