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A Monte Carlo estimation of the entropy for Markov chains

Abstract : We introduce an estimate of the entropy $\bE_{p^t}(\log p^t)$ of the marginal density $p^t$ of a (eventually inhomogeneous) Markov chain at time~$t\ge1$. 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 {\it external} entropy $\bE_{p_1^t}(\log p^t)$, 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: (i) a diagnostic by simulation of the stability property of a Markovian dynamical system with respect to various initial conditions; (ii) a study of the rate in the Central Limit Theorem for i.i.d.\ random variables. Simulated examples are provided as illustration.
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https://hal.archives-ouvertes.fr/hal-00140942
Contributor : Didier Chauveau <>
Submitted on : Tuesday, April 10, 2007 - 7:24:48 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

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Didier Chauveau, Pierre Vandekerkhove. A Monte Carlo estimation of the entropy for Markov chains. Methodology and Computing in Applied Probability, Springer Verlag, 2007, 9 (1), pp.133-149. ⟨10.1007/s11009-006-9010-6⟩. ⟨hal-00140942⟩

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