How to compare MCMC simulation strategies?

Abstract. In MCMC methods, such as the Metropolis-Hastings (MH) algorithm, the Gibbs sampler, or recent adaptive methods, many different strategies can be proposed, often asso-ciated in practice to unknown rates of convergence. In this paper we propose a simulation-based methodology to compare these rates of convergence, grounded on an entropy criterion computed from parallel (i.i.d.) simulated Markov chains coming from each candidate strat-egy. Our criterion determines the most efficient strategy among the candidates. Theoreti-cally, we give for the MH algorithm general conditions under which its successive densities satisfy adequate smoothness and tail properties, so that this entropy criterion can be esti-mated consistently using kernel density estimate and Monte Carlo integration. Simulated and actual examples in moderate dimensions are provided to illustrate this approach