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Autour du maximum d'entropie et de l'information de Rényi
28-févr.-2007 14:25 14:25
Il y a: 14 yrs


AUTEUR : JEAN-FRANÇOIS BERCHER



Autour du maximum d'entropie et de l'information de Rényi


Séminaire Scientifique  « Recherche »

Présentation des travaux de recherche

Jean-François BERCHER

Jeudi 08 Février 2007, Amphi 110 à l’Esiee

- De 13h30 à 14h30 : autour du maximum d'entropie, et de l'information de Rényi

mots-clé : Entropies, Rényi, physique statistique, non-extensivité, convexité

 

De 13h30 à 14h30 : autour du maximum d'entropie, et de l'information de Rényi

An amended MaxEnt formulation for deriving Tsallis factors, and associated issues

An amended MaxEnt formulation for systems displaced from the conventional MaxEnt equilibrium is proposed. This formulation involves the minimization of the Kullback-Leibler divergence to a reference Q (or maximization of Shannon Q-entropy), subject to a constraint that implicates a second reference distribution P_{1} and tunes the new equilibrium. In this setting, the equilibrium distribution is the generalized escort distribution associated to P_{1} and Q. The account of an additional constraint, an observable given by a statistical mean, leads to the maximization of Rényi/Tsallis Q-entropy subject to that constraint. Two natural scenarii for this observation constraint are considered, and the classical and generalized constraint of nonextensive statistics are recovered. The solutions to the maximization of Rényi Q-entropy subject to the two types of constraints are derived. These optimum distributions, that are Levy-like distributions, are self-referential. We then propose two  `alternate' (but effectively computable) dual functions, whose maximizations enable to identify the optimum parameters. Finally, a duality between solutions and the underlying Legendre structure are presented.

On Some Entropy Functionals derived from Rényi Information Divergence

I consider the notion of entropy functionals in the object space, {F}_{\alpha}^{(1)(x) and {F}_{\alpha}^{(\alpha)}(x), that correspond to the minimization of Rényi divergence  to Q subject to the two kinds of observation constraints. I relate these functionals to the partition function of the associated distributions, characterize their main properties and behaviors, indicate an underlying symmetry and introduce a divergence in the object space.

I examine two special cases of reference Q(x), in the continuous and discrete case: an exponential measure and a Poisson distribution. In each case, I derive the expressions of the partition function and of its derivative, and give the expressions of the entropy functionals {F}_{\alpha}^{(.)}(x)  for \alpha-->1. I also present results of numerical evaluations for \alpha\in [0,1].

 

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