New Metric and Connections in Statistical Manifolds

Abstract

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).

Geometry Induced by a Generalization of Rényi Divergence

“…In [17,18], the authors introduced a pair of dual connections D (−1) and D on how the generalization of Rényi divergence is related to Zhang's (ρ, τ)-divergence, and also how the present proposal is related to the model presented in [33].…”

“…Instead of the logarithm, we consider the inverse ϕ −1 (·) of a ϕ-function. Generalized statistical manifolds were introduced by the authors in [17,18]. Among examples of the generalized statistical manifold, (parametric) ϕ-families of probability distributions are of greatest importance.…”

“…The metric in ϕ-families can be defined as the Hessian of a function; i.e., ϕ-families are Hessian manifolds [30]. In [17,18] …”

“…Based on the similarity between exponential and ϕ-families, we defined the so-called ϕ-divergence, with respect to which the Kullback-Leibler divergence is a particular case. Statistical models equipped with a geometric structure induced by ϕ-divergences, which are called generalized statistical manifolds, are investigated in [17,18]. With respect to these connections, parametric ϕ-families are dually flat.…”

“…In a limiting case, the generalized Rényi divergence reduces to the ϕ-divergence. In [17,18], the ϕ-divergence gives rise to a pair of dual connections D (−1) and D (1) . We show that the connection D (α) induced by the generalization of Rényi divergence satisfies the convex combination [26] investigated a geometry based on a normalizing function similar to the one used in the generalization of Rényi divergence.…”