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August 2013 Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition
Alicia A. Johnson, Galin L. Jones, Ronald C. Neath
Statist. Sci. 28(3): 360-375 (August 2013). DOI: 10.1214/13-STS423

Abstract

It is common practice in Markov chain Monte Carlo to update the simulation one variable (or sub-block of variables) at a time, rather than conduct a single full-dimensional update. When it is possible to draw from each full-conditional distribution associated with the target this is just a Gibbs sampler. Often at least one of the Gibbs updates is replaced with a Metropolis–Hastings step, yielding a Metropolis–Hastings-within-Gibbs algorithm. Strategies for combining component-wise updates include composition, random sequence and random scans. While these strategies can ease MCMC implementation and produce superior empirical performance compared to full-dimensional updates, the theoretical convergence properties of the associated Markov chains have received limited attention. We present conditions under which some component-wise Markov chains converge to the stationary distribution at a geometric rate. We pay particular attention to the connections between the convergence rates of the various component-wise strategies. This is important since it ensures the existence of tools that an MCMC practitioner can use to be as confident in the simulation results as if they were based on independent and identically distributed samples. We illustrate our results in two examples including a hierarchical linear mixed model and one involving maximum likelihood estimation for mixed models.

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Alicia A. Johnson. Galin L. Jones. Ronald C. Neath. "Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition." Statist. Sci. 28 (3) 360 - 375, August 2013. https://doi.org/10.1214/13-STS423

Information

Published: August 2013
First available in Project Euclid: 28 August 2013

zbMATH: 1331.60151
MathSciNet: MR3135537
Digital Object Identifier: 10.1214/13-STS423

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.28 • No. 3 • August 2013
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