Rates of convergence of the Hastings and Metropolis algorithms

The Annals of Statistics · 1996 · Volume 24 · Issue 1

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.

Keywords

62-04; 62J05; 65C05; geometric ergodicity; Gibbs sampling; Hastings algorithms; irreducible Markov processes; log-concave distributions; Markov chain Monte Carlo; Metropolis algorithms; posterior distributions; Stochastic monotonicity;

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