Exploring information geometry: Recent Advances and Connections to Topological Field Theory

TL;DR

This paper introduces the interplay between differential geometry, probability theory, and algebraic structures like Frobenius manifolds, illustrating their role in the geometry of information for a broad mathematical audience.

Contribution

It synthesizes ideas connecting geometry and statistics, emphasizing the emergence of geometric structures in statistical models and their relation to topological field theory.

Findings

Connection between exponential families and Frobenius manifolds

Geometric interpretation of statistical models

Introduction to the role of curvature and connections in information geometry

Abstract

This introductory text arises from a lecture given in G\"oteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas connecting geometry and statistics. At its core, this work seeks to illustrate the profound and yet natural interplay between differential geometry, probability theory, and the rich algebraic structures encoded in (pre-)Frobenius manifolds. The exposition is structured into three principal parts. The first part provides a concise introduction to differential topology and geometry, emphasizing the role of smooth manifolds, connections, and curvature in the formulation of geometric structures. The second part is devoted to probability, measures, and statistics, where the notion of a probability space is refined into a geometric object, thus paving the way for a deeper mathematical understanding of statistical models. Finally, in the third part, we introduce (pre-)Frobenius manifolds, revealing their surprising connection to exponential families of probability distributions and, discuss more broadly, their role in the geometry of information. At the end of those three parts the reader will find stimulating exercises. By bringing together these seemingly distant disciplines, we aim to highlight the natural emergence of geometric structures in statistical theory. This work does not seek to be exhaustive but rather to provide the reader with a pathway into a domain of mathematics that is still in its formative stages, where many fundamental questions remain open. The text is accessible without requiring advanced prerequisites and should serve as an invitation to further exploration.