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

• 成果类型: Article
• 署名作者: Roberts, Gareth O.; Rosenthal, Jeffrey S.
• 署名单位: Lancaster University; University of Toronto
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000510
• 发表日期: 2006
• 页码: 2123-2139
• 关键词: exploring posterior distributions; monte-carlo; convergence-rates

摘要

A phi-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.