Improving convergence of the Hastings - MetropolisAlgorithm with a learning proposal

The Hastings-Metropolis algorithm is a general MCMC method for sampling from a density known up to a constant. Geometric convergence of this algorithm has been proved under conditions relative to the instrumental distribution (or proposal). We present an inhomogeneous Hastings-Metropolis algorithm for which the proposal density approximates the target density, as the number of iterations increases. The proposal at the nth step is a nonparametric estimate of the density of the algorithm, and uses an increasing number of iid copies of the Markov chain. The resulting algorithm converges (in n) geometrically faster than a Hastings-Metropolis algorithm with an arbitrary proposal. The case of a strictly positive density with compact support is presented rst, then an extension to more general densities is given. We conclude by proposing a practical way of implementation for the algorithm, and illustrate it over a simulated example.