Sufficient burn-in for Gibbs samplers for a hierarchical random effects model

By Galin L. Jones, James P. Hobert. 2004

117 citations recorded by the source corpus.

We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical version of the one-way random effects model. Drift and minorization conditions are established for the underlying Markov chains. The drift and minorization are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558– 566] and G. O. Roberts and R. L. Tweedie [Stochastic Process. Appl. 80 (1999) 211–229] to construct analytical upper bounds on the distance to stationarity. These lead to upper bounds on the amount of burn-in that is required to get the chain within a prespecified (total variation) distance of the stationary distribution. The results are illustrated with a numerical example. 1. Introduction. We consider a Bayesian hierarchical version of the standard normal theory one-way random effects model. The posterior density for this model is intractable in the sense that the integrals required for making inferences cannot be computed in closed form. Hobert and Geyer (1998) analyzed a Gibbs sampler and a block Gibbs sampler for this problem and showed that the Markov chains underlying these algorithms converge to the stationary (i.e., posterior) distribution at a geometric rate. However, Hobert and Geyer stopped short of constructing analytical upper bounds on the total variation distance to stationarity. In this article, we construct such upper bounds and this leads to a method for determining a sufficient burn-in. Our results are useful from a practical standpoint because they obviate troublesome, ad hoc convergence diagnostics [Cowles and Carlin (1996) and

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d447b5e529bd3332646aed1aa00faf8ca8531098

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Citations in the corpus

Multivariate output analysis for Markov chain Monte Carlo (2015, background): “The condition is fairly weak since often the existence of the Markov chain CLT itself is demonstrated via polynomial ergodicity or a stronger result (see Jones (2004) for a review).”
Gibbs sampling, exponential families and orthogonal polynomials (2008, methodology): “M E ] 2 8 A ug 2 00 8 Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition.”
Markov chain Monte Carlo: Can we trust the third significant figure? (2007, background): “For example, Hobert and Geyer (1998), Johnson and Jones (2008), Jones and Hobert (2004), Marchev and Hobert (2004), Mira and Tierney (2002), Robert (1995), Roberts and Polson (1994), Roberts and Rosenthal (1999), Rosenthal (1995, 1996), Roy and Hobert (2007) and Tierney (1994) examined Gibbs…”
Batch means and spectral variance estimators in Markov chain Monte Carlo (2008, background): “Roy and Hobert [47] prove that this sampler is geometrically ergodic.”
On the Markov chain central limit theorem (2004, background): “Several m otivating exam ples are given which range from toy one-dim ensionalsettings to com plicated settingsencountered in M arkov chain M onte Carlo.”
General state space Markov chains and MCMC algorithms (2004, background): “And, how large is large enough?”
Fixed-Width Output Analysis for Markov Chain Monte Carlo (2006, background): “Sequential statistical procedures have a long history (see Lai 2001 for an overview and commentary).”