Exploring information geometry: Recent Advances and Connections to Topological Field Theory
Noémie C. Combe, Philippe G. Combe, Hanna K. Nencka
TL;DR
This work surveys the synthesis of geometry and probability through information geometry, tracing a path from general topology to differentiable manifolds, fiber bundles, and the geometric formulation of probability spaces. It articulates three integrated pillars: (i) topology and geometry of manifolds and bundles, (ii) probabilistic structures grounded in measure theory and statistics, and (iii) Frobenius-manifold formalism linking exponential families to information geometry and topological field theory via WDVV equations. By detailing connections between connections, curvature, and statistical models, it highlights how coordinate choices can reveal deep algebraic and geometric structures underlying probabilistic theories. The text is designed as an accessible entry point with exercises, aimed at students and researchers seeking foundational and cutting-edge insights into the geometry of information. The overall significance lies in establishing a rigorous geometric framework for probability distributions and their applications in physics, statistics, and data analysis.
Abstract
This introductory text arises from a lecture given in Göteborg, 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.