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A Monte Carlo Estimation of the Entropy for Markov Chains

Author

Listed:
  • Didier Chauveau

    (Université d’Orléans)

  • Pierre Vandekerkhove

    (Université de Marne-la-Vallée)

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.

Suggested Citation

  • Didier Chauveau & Pierre Vandekerkhove, 2007. "A Monte Carlo Estimation of the Entropy for Markov Chains," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 133-149, March.
    • Handle: RePEc:spr:metcap:v:9:y:2007:i:1:d:10.1007_s11009-006-9010-6
    DOI: 10.1007/s11009-006-9010-6
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    1. Munsup Seoh & Marc Hallin, 1997. "When does Edgeworth beat Berry and Esséen? Numerical evaluations of Edgeworth expansions," ULB Institutional Repository 2013/127957, ULB -- Universite Libre de Bruxelles.
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