Open Access
February 1996 Rates of convergence of the Hastings and Metropolis algorithms
K. L. Mengersen, R. L. Tweedie
Ann. Statist. 24(1): 101-121 (February 1996). DOI: 10.1214/aos/1033066201


We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution $\pi$. In the independence case (in $\mathbb{R}^k$) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of $\pi$; in the symmetric case (in $\mathbb{R}$ only) we show geometric convergence essentially occurs if and only if $\pi$ has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective.


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K. L. Mengersen. R. L. Tweedie. "Rates of convergence of the Hastings and Metropolis algorithms." Ann. Statist. 24 (1) 101 - 121, February 1996.


Published: February 1996
First available in Project Euclid: 26 September 2002

zbMATH: 0854.60065
MathSciNet: MR1389882
Digital Object Identifier: 10.1214/aos/1033066201

Primary: 62-04 , 62J05 , 65C05

Keywords: geometric ergodicity , Gibbs sampling , Hastings algorithms , irreducible Markov processes , log-concave distributions , Markov chain Monte Carlo , Metropolis algorithms , posterior distributions , Stochastic monotonicity

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 1 • February 1996
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