HAL : hal-00276749, version 1

arXiv : 0805.0129

DOI : 10.1016/j.ins.2008.02.003

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Information Sciences 178, 12 (2008) 2489-2506

On some entropy functionals derived from Rényi information divergence

Jean-François Bercher 1, 2

(15/06/2008)

We consider the maximum entropy problems associated with Rényi $Q$-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized. The Rényi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure $Q(x)$ and recover in a limit case some well-known entropies.

1 :	Laboratoire des signaux et systèmes (L2S)
	UMR8506 CNRS – SUPELEC – Université Paris XI - Paris Sud
2 :	Laboratoire d'Informatique Gaspard-Monge (LIGM)
	Université Paris-Est Marne-la-Vallée (UPEMLV) – ESIEE – Ecole des Ponts ParisTech – Fédération de Recherche Bézout – CNRS : UMR8049

Domaine

:

Informatique/Théorie de l'information et codage Mathématiques/Théorie de l'information et codage Physique/Matière Condensée/Autre

Mots Clés : Maximum entropy principle – Nonextensivity – Rényi divergences – Rényi entropy – Tsallis distributions

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Contributeur : Jean-François Bercher <>

Soumis le : Jeudi 1 Mai 2008, 19:37:11

Dernière modification le : Jeudi 1 Mai 2008, 21:09:21