Information geometry derived from divergence functions

Given a divergence function in a manifold, we can derive a Riemannian metric and a dual pair of affine connections, which are the essential constituents of Information Geometry. In the case of a family of probability distributions, its information-geometric structure is given from the invariance property. It consists of the Fisher information metric and the alpha connections. Moreover, the manifold of discrete probability distributions, that is the set of all probability distribution on a finite set, has a dually flat Riemannian structure. We study how these properties are related to the underlying divergence function. We define an invariant divergence in terms of information monotonicity, which leads us to the class of $f$-divergences. We then study a divergence function which gives dually flat affine structure. This is given by the Bregman divergence in terms of a convex function. The invariant and flat divergence in the manifold of probability distributions is the Kullback-Leibler divergence, and this is unique, but more generally it is the class of alpha-divergences in the manifold of positive measures. We can further discuss divergence functions in the manifold of positive-definite matrices, that of vision pictures and cones. A nonlinear transformation of a divergence function or a convex function causes a conformal change of the dual geometrical structure. In this context, we can discuss the dual geometry derived from the Tsallis or Renyi entropy. It again gives the dually flat structure to the family of discrete probability distributions or of positive measures. This can be extended to the family of positive-definite matrices. We can define the $q$-exponential family and related $q$-structure, which is a generalization of the current invariant information geometry.