Open Access
2010 Gibbs sampling for a Bayesian hierarchical general linear model
Alicia A. Johnson, Galin L. Jones
Electron. J. Statist. 4: 313-333 (2010). DOI: 10.1214/09-EJS515


We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergodicity is also a key requirement for using batch means methods to consistently estimate the variance of the asymptotic normal distribution. Together, our results give practitioners the tools to be as confident in inferences based on the observations from the Gibbs sampler as they would be with inferences based on random samples from the posterior. Our theoretical results are illustrated with an application to data on the cost of health plans issued by health maintenance organizations.


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Alicia A. Johnson. Galin L. Jones. "Gibbs sampling for a Bayesian hierarchical general linear model." Electron. J. Statist. 4 313 - 333, 2010.


Published: 2010
First available in Project Euclid: 15 March 2010

zbMATH: 1329.62336
MathSciNet: MR2645487
Digital Object Identifier: 10.1214/09-EJS515

Keywords: convergence rate , drift condition , General linear model , geometric ergodicity , Gibbs sampler , Markov chain , Monte Carlo

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

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