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Open AccessArticle

Geometry Induced by a Generalization of Rényi Divergence

Instituto Federal do Ceará, Campus Maracanaú, Fortaleza 61939-140, Brazil
Computer Engineering School, Campus Sobral, Federal University of Ceará, Sobral 62010-560, Brazil
Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza 60455-900, Brazil
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Entropy 2016, 18(11), 407;
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)
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
MDPI and ACS Style

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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