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A Monte Carlo Estimation of the Entropy for Markov Chains

  • Didier ChauveauEmail author
  • Pierre Vandekerkhove
Article

Abstract

We introduce an estimate of the entropy \(\mathbb{E}_{p^t}(\log p^t)\) of the marginal density p t of a (eventually inhomogeneous) Markov chain at time t≥1. This estimate is based on a double Monte Carlo integration over simulated i.i.d. copies of the Markov chain, whose transition density kernel is supposed to be known. The technique is extended to compute the external entropy \(\mathbb{E}_{p_1^t}(\log p^t)\), where the p 1 t s are the successive marginal densities of another Markov process at time t. We prove, under mild conditions, weak consistency and asymptotic normality of both estimators. The strong consistency is also obtained under stronger assumptions. These estimators can be used to study by simulation the convergence of p t to its stationary distribution. Potential applications for this work are presented: (1) a diagnostic by simulation of the stability property of a Markovian dynamical system with respect to various initial conditions; (2) a study of the rate in the Central Limit Theorem for i.i.d. random variables. Simulated examples are provided as illustration.

Keywords

Entropy Kullback information Markov chains Stability Rate in the Central Limit Theorem 

AMS 2000 Subject Classification

62B10 60F05 60J10 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Laboratoire MAPMO-UMR 6628—Fédération Denis PoissonUniversité d’OrléansOrléans Cedex 2France
  2. 2.CNRS UMR 8050Université de Marne-la-ValléeMarne-la-ValléeFrance

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