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Entropy 2016, 18(11), 407; doi:10.3390/e18110407

Geometry Induced by a Generalization of Rényi Divergence

1
Instituto Federal do Ceará, Campus Maracanaú, Fortaleza 61939-140, Brazil
2
Computer Engineering School, Campus Sobral, Federal University of Ceará, Sobral 62010-560, Brazil
3
Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza 60455-900, Brazil
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Received: 6 September 2016 / Revised: 27 October 2016 / Accepted: 11 November 2016 / Published: 17 November 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
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Abstract

In this paper, we propose a generalization of Rényi divergence, and then we investigate its induced geometry. This generalization is given in terms of a φ-function, the same function that is used in the definition of non-parametric φ-families. The properties of φ-functions proved to be crucial in the generalization of Rényi divergence. Assuming appropriate conditions, we verify that the generalized Rényi divergence reduces, in a limiting case, to the φ-divergence. In generalized statistical manifold, the φ-divergence induces a pair of dual connections D ( 1 ) and D ( 1 ) . We show that the family of connections D ( α ) induced by the generalization of Rényi divergence satisfies the relation D ( α ) = 1 α 2 D ( 1 ) + 1 + α 2 D ( 1 ) , with α [ 1 , 1 ] . View Full-Text
Keywords: Rényi divergence; φ-function; φ-divergence; φ-family; statistical manifold; information geometry Rényi divergence; φ-function; φ-divergence; φ-family; statistical manifold; information geometry
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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de Souza, D.C.; Vigelis, R.F.; Cavalcante, C.C. Geometry Induced by a Generalization of Rényi Divergence. Entropy 2016, 18, 407.

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