Smoothness of Metropolis-Hastings algorithm and application to entropy estimation

¹ MAPMO – UMR 7349, Fédération Denis Poisson, Université d’Orléans et CNRS BP 6759, 45067 Orléans Cedex 2, France
didier.chauveau@univ-orleans.fr
² LAMA, CNRS UMR 8050, Université de Marne-la-Vallée, 5 Bd. Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
Pierre.Vandekerkhove@univ-mlv.fr

Received: 27 April 2011
Revised: 6 February 2012

Abstract

The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain’s current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm. We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved. This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities. This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates.