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November 2006 Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains
Gareth O. Roberts, Jeffrey S. Rosenthal
Ann. Appl. Probab. 16(4): 2123-2139 (November 2006). DOI: 10.1214/105051606000000510


A ϕ-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace “almost all” by “all,” which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis–Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional counter-example showing how Harris recurrence can fail, and also a variety of positive results which guarantee Harris recurrence. We also present some open problems. We close with a discussion of the practical implications for MCMC algorithms.


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Gareth O. Roberts. Jeffrey S. Rosenthal. "Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains." Ann. Appl. Probab. 16 (4) 2123 - 2139, November 2006.


Published: November 2006
First available in Project Euclid: 17 January 2007

zbMATH: 1121.60076
MathSciNet: MR2288716
Digital Object Identifier: 10.1214/105051606000000510

Primary: 60J05
Secondary: 60J22 , 62F15 , 65C05

Keywords: Harris recurrence , Markov chain Monte Carlo , Metropolis algorithm , phi-irreducibility , trans-dimensional Markov chains

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.16 • No. 4 • November 2006
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