The Annals of Statistics

Rates of convergence of the Hastings and Metropolis algorithms

K. L. Mengersen and R. L. Tweedie

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 26 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1033066201

Digital Object Identifier
doi:10.1214/aos/1033066201

Mathematical Reviews number (MathSciNet)
MR1389882

Zentralblatt MATH identifier
0854.60065

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

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

Citation

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. http://projecteuclid.org/euclid.aos/1033066201.


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