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Smoothness of Metropolis-Hastings algorithm and application to entropy estimation

Published online by Cambridge University Press:  21 May 2013

Didier Chauveau  and
Pierre Vandekerkhove
Didier Chauveau
Affiliation:
MAPMO – UMR 7349, Fédération Denis Poisson, Université d’Orléans et CNRS BP 6759, 45067 Orléans Cedex 2, France. didier.chauveau@univ-orleans.fr
Pierre Vandekerkhove
Affiliation:
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
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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.

Keywords

EntropyKullback divergenceMetropolis-Hastings algorithmnonparametric statistic
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 419 - 431
Copyright
© EDP Sciences, SMAI, 2013

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