Dually affine Information Geometry modeled on a Banach space

TL;DR

This paper explores a non-parametric approach to Information Geometry by modeling probability measures within a Banach space framework, emphasizing affine transition mappings and manifold structures.

Contribution

It introduces a novel affine manifold structure for probability measures in a Banach space, focusing on non-parametric and functional perspectives in Information Geometry.

Findings

Constructed a manifold structure with affine transition maps

Defined tangent and cotangent bundles in the affine setting

Provided a non-parametric geometric framework for probability measures

Abstract

In this chapter, we study Information Geometry from a particular non-parametric or functional point of view. The basic model is a probabilities subset usually specified by regularity conditions. For example, probability measures mutually absolutely continuous or probability densities with a given degree of smoothness. We construct a manifold structure by giving an atlas of charts as mappings from probabilities to a Banach space. The charts we use are quite peculiar in that we consider only instances where the transition mappings are affine. We chose a particular expression of the tangent and cotangent bundles in this affine setting.