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

We define a metric and a family of α-connections in statistical manifolds, based on ϕ-divergence, which emerges in the framework of ϕ-families of probability distributions. This metric and α-connections generalize the Fisher information metric and Amari's α-connections. We also investigate the parallel transport associated with the α-connection for α = 1. IntroductionIn the framework of ϕ-families of probability distributions [11], the authors introduced a divergence D ϕ (· ·) between probabilities distributions, called ϕ-divergence, that generalizes the Kullback-Leibler divergence. Based on D ϕ (· ·) we can define a new metric and connections in statistical manifolds. The definition of metrics or connections in statistical manifolds is a common subject in the literature [2,3,7]. In our approach, the metric and α-connections are intrinsically related to ϕ-families. Moreover, they can be recognized as a generalization of the Fisher information metric and Amari's α-connections [1,4].Statistical manifolds are equipped with the Fisher information metric, which is given in terms of the derivative of l(t; θ) = log p(t; θ). Another metric can be defined if the logarithm log(·) is replaced by the inverse of a ϕ-function ϕ(·) [11]. Instead of l(t; θ) = log p(t; θ), we can consider f (t; θ) = ϕ −1 (p(t; θ)). The manifold equipped with * This work was partially funded by CNPq (Proc. 309055/2014-8).

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