Rates of convergence of the Hastings and Metropolis algorithms

Mengersen, K. L. ; Tweedie, R. L.

Ann. Statist., Tome 24 (1996) no. 6, p. 101-121 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1996-02-14
Classification:  Posterior distributions,  Hastings algorithms,  Metropolis algorithms,  Gibbs sampling,  Markov chain Monte Carlo,  irreducible Markov processes,  geometric ergodicity,  stochastic monotonicity,  log-concave distributions,  62J05,  62-04,  65C05