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Entropy 2018, 20(6), 436; https://doi.org/10.3390/e20060436

Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

1
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, 10129 Torino, Italy
2
Department of Computer Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan
3
Region of Electrical and Electronic Systems Engineering, Ibaraki University, Nakanarusawa-cho, Hitachi 316-8511, Japan
*
Author to whom correspondence should be addressed.
Received: 9 May 2018 / Revised: 31 May 2018 / Accepted: 1 June 2018 / Published: 5 June 2018
(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)
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Abstract

In this paper, we present a review of recent developments on the κ -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ -formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the κ -deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the κ -Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the κ 0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the κ -escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of κ -thermodynamics in the picture of the information geometry. View Full-Text
Keywords: κ-generalized statistical mechanics; information geometry; dually-flat geometry; Hessian geometry; Legendre structure; divergence functions κ-generalized statistical mechanics; information geometry; dually-flat geometry; Hessian geometry; Legendre structure; divergence functions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Scarfone, A.M.; Matsuzoe, H.; Wada, T. Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures. Entropy 2018, 20, 436.

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