Geometry Induced by a Generalization of Rényi Divergence

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 ] .