Statistical manifolds and affine differential geometry

Abstract

In this paper, we give a summary of geometry of statistical manifolds, and discuss relations between information geometry and affine differential geometry. Dually flat spaces and canonical divergence functions are important objects in information geometry. We show that such objects can be generalized in the framework of affine differential geometry.

In addition, we give a brief summary of geometry of statistical manifolds admitting torsion, which is regarded as a quantum version of statistical manifolds. We discuss relations between statistical manifolds admitting torsion and geometry of affine distributions.