Information geometry of the space of probability measures and barycenter maps

TL;DR

This paper explores the geometric structure of probability measures using information geometry, focusing on Fisher metric geodesics and barycenters on Hadamard manifolds, revealing new insights into their mathematical properties.

Contribution

It introduces recent developments in the geometry of probability measures, including Fisher metric geodesics and barycenter theory on Hadamard manifolds, advancing the understanding of their geometric and probabilistic relationships.

Findings

Analysis of Fisher metric geodesics

Properties of barycenters on Hadamard manifolds

Connections between information geometry and probability measures

Abstract

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.