Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries

Shun-ichi Amari, Atsumi Ohara and Hiroshi Matsuzoe

Physica A: Statistical Mechanics and its Applications, 2012, vol. 391, issue 18, 4308-4319

Abstract: An information-geometrical foundation is established for the deformed exponential families of probability distributions. Two different types of geometrical structures, an invariant geometry and a flat geometry, are given to a manifold of a deformed exponential family. The two different geometries provide respective quantities such as deformed free energies, entropies and divergences. The class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families. Theq-families are characterized from the viewpoint of the invariant and flat geometries. The q-exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures. Furthermore, it is the only class of which the Riemannian metric is conformally connected with the invariant Fisher metric.

Keywords: Generalized entropies; Deformed exponential families; Information geometry; Invariance principle; Conformal transformation (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (6)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:391:y:2012:i:18:p:4308-4319

DOI: 10.1016/j.physa.2012.04.016

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