Improving convergence of the Hastings-Metropolis algorithm with an adaptive 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 (or proposal) distribution. We present an inhomogeneous Hastings-Metropolis algorithm for which the proposal density approximates the target density, as the number of iterations increases. The proposal density at the nth step is a non-parametric estimate of the density of the algorithm, and uses an increasing number of i.i.d. copies of the Markov chain. The resulting algorithm converges (in n) geometrically faster than a Hastings-Metropolis algorithm with any fixed proposal distribution. The case of a strictly positive density with compact support is presented first, 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 simulated examples.