The Annals of Statistics

Rates of convergence of the Hastings and Metropolis algorithms

K. L. Mengersen and R. L. Tweedie

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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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Ann. Statist., Volume 24, Number 1 (1996), 101-121.

First available in Project Euclid: 26 September 2002

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Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62-04: Explicit machine computation and programs (not the theory of computation or programming) 65C05: Monte Carlo methods

Posterior distributions Hastings algorithms Metropolis algorithms Gibbs sampling Markov chain Monte Carlo irreducible Markov processes geometric ergodicity stochastic monotonicity log-concave distributions


Mengersen, K. L.; Tweedie, R. L. Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 (1996), no. 1, 101--121. doi:10.1214/aos/1033066201.

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