Statistical Manifolds Admitting Torsion and Partially Flat Spaces

Abstract

It is well-known that a contrast function defined on a product manifold $M \times M$ induces a Riemannian metric and a pair of dual torsion-free affine connections on the manifold M. This geometrical structure is called a statistical manifold and plays a central role in information geometry. Recently, the notion of pre-contrast function has been introduced and shown to induce a similar differential geometrical structure on M, but one of the two dual affine connections is not necessarily torsion-free. This structure is called a statistical manifold admitting torsion. The notion of statistical manifolds admitting torsion has been originally introduced to study a geometrical structure which appears in a quantum statistical model. However, it has been shown that an estimating function which is used in “classical” statistics also induces a statistical manifold admitting torsion through its associated pre-contrast function. The aim of this paper is to summarize such previous results. In particular, we focus on a partially flat space, which is a statistical manifold admitting torsion where one of its dual connections is flat. In this space, it is possible to discuss some properties similar to those in a dually flat space, such as a canonical pre-contrast function and a generalized projection theorem.