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Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains

Gareth O. Roberts·Jeffrey S. Rosenthal
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Abstract

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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BibTeXRIS

Gareth O. Roberts, Jeffrey S. Rosenthal. 2007-02-14. Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. https://doi.org/10.1214/105051606000000510

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