Geometry Induced by a Generalization of Renyi Divergence

In this paper, we propose a generalization of Renyi divergence, and then we investigate its induced geometry. This generalization is given in terms of a phi-function, the same function that is used in the definition of non-parametric phi-families. The properties of phi-functions proved to be crucial in the generalization of Renyi divergence. Assuming appropriate conditions, we verify that the generalized Renyi divergence reduces, in a limiting case, to the phi-divergence. In generalized statistical manifold, the phi-divergence induces a pair of dual connections relation D (a) = 1 a 2 D (1) + 1 + a 2 D (1), with alpha is an element of [-1, 1].