Generalized information-entropy measures and Fisher information

Abstract

We show how Fisher's information already known particular character as the fundamental information geometric object which plays the role of a metric tensor for a statistical differential manifold, can be derived in a relatively easy manner through the direct application of a generalized logarithm and exponential formalism to generalized information-entropy measures. We shall first shortly describe how the generalization of information-entropy measures naturally comes into being if this formalism is employed and recall how the relation between all the information measures is best understood when described in terms of a particular logarithmic Kolmogorov-Nagumo average. Subsequently, extending Kullback-Leibler's relative entropy to all these measures defined on a manifold of parametrized probability density functions, we obtain the metric which turns out to be the Fisher information matrix elements times a real multiplicative deformation parameter. The metrics independence from the non-extensive character of the system, and its proportionality to the rate of change of the multiplicity under a variation of the statistical probability parameter space, emerges naturally in the frame of this representation.

16 pages, 1 diagram (some references added)

References in corpus (6)

• Geometry of escort distributions
• Estimators, escort probabilities, and phi-exponential families in statistical physics
• Escort--Husimi distributions, Fisher information and nonextensivity
• Geometrical aspects of a generalized statistical mechanics
• Fluctuations, gravity, and the quantum potential
• Generalized thermostatistics and Kolmogorov-Nagumo averages

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• On the extended Kolmogorov-Nagumo information-entropy theory, the q -> 1/q duality and its possible implications for a non-extensive two dimensional Ising model
• Towards an Information Geometric characterization/classification of Complex Systems. I. Use of Generalized Entropies
• Dynamics of Information Entropies in Nonextensive Systems
• Generalized information entropies in nonextensive quantum systems: The interpolation approach