Geometry Induced by a Generalization of Rényi Divergence

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].