Computation and Estimation of Generalized Entropy Rates for Denumerable Markov Chains

Gabriela Ciuperca (1) , Valérie Girardin (2) , Loïck Lhote (3)

1 LAPCS - Laboratoire de Probabilités, Combinatoire et Statistique
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 Equipe AMACC - Laboratoire GREYC - UMR6072

—We study entropy rates of random sequences for general entropy functionals including the classical Shannon and Rényi entropies and the more recent Tsallis and Sharma-Mittal ones. In the first part, we obtain an explicit formula for the entropy rate for a large class of entropy functionals, as soon as the process satisfies a regularity property known in dynamical systems theory as the quasi-power property. Independent and identically distributed sequence of random variables naturally satisfy this property. Markov chains are proven to satisfy it too, under simple explicit conditions on their transition probabilities. All the entropy rates under study are thus shown to be either infinite or zero except at a threshold where they are equal to Shannon or Rényi entropy rates up to a multiplicative constant. In the second part, we focus on the estimation of the marginal generalized entropy and entropy rate for parametric Markov chains. Estimators with good asymptotic properties are built through a plug-in procedure using a maximum likelihood es-timation of the parameter.

Mots clés: Rényi entropy; Tsallis entropy; plug-in estimation; Index Terms—entropy rate; entropy functional; parametric Markov chain

Domaines: Algorithme et structure de données [cs.DS]; Complexité [cs.CC]; Théorie de l'information [cs.IT]

Gabriela Ciuperca, Valérie Girardin, Loïck Lhote. Computation and Estimation of Generalized Entropy Rates for Denumerable Markov Chains. IEEE Transactions on Information Theory, 2011, 57, pp.4026 - 4034. ⟨10.1109/TIT.2011.2133710⟩. ⟨hal-01082088⟩