Geometry Induced by a Generalization of Rényi Divergence

David de Souza, Rui Vigelis, Charles Cavalcante
2016 Entropy  
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
more » ... zed 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].
doi:10.3390/e18110407 fatcat:btvmgznudzfeho5kaezgqjg67y