RATES OF CONVERGENCE OF THE HASTINGS AND METROPOLIS ALGORITHMS

We apply recent results in Markov chain theory to Hastings and Metropo lis algorithms with either independent or symmetric candidate distribu tions, and provide necessary and sufficient conditions for the algorit hms to converge at a geometric rate to a prescribed distribution pi. I n the independence case (in R(k)) these indicate that geometric conver gence essentially occurs if and only if the candidate density is bound ed below by a multiple of pi; in the symmetric case (in R only) we sho w geometric convergence essentially occurs if and only if pi has geome tric tails. We also evaluate recently developed computable bounds on t he rates of convergence in this context: examples show that these theo retical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective .