Construction of Exponential Families from Statistical Manifolds

TL;DR

This paper explores how statistical manifolds can be decomposed into exponential families using geometric and topological methods, revealing deep links between information geometry and 3-manifold topology.

Contribution

It provides a constructive method to foliate compact statistical manifolds into Hessian leaves, characterizes their geometric structures, and connects them explicitly to exponential families.

Findings

Every compact statistical manifold admits a foliation into Hessian manifolds.

Non-flat, compact, orientable 3D leaves are quotients of exponential families with odd Betti numbers.

Non-flat 3D leaves have co-Kähler structures and can be realized as exponential families parametrized by Lorentz cones.

Abstract

We investigate the construction of exponential families from statistical manifolds, a central problem in information geometry. We prove that every compact statistical manifold admits a singular foliation whose leaves are Hessian manifolds. In particular, any non-flat, compact, orientable 3-dimensional leaf arises as a quotient of an exponential family and has only odd Betti numbers. Our approach is constructive: we explicitly describe the foliation and analyze the geometric and topological properties of its leaves. We show that compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy. In three dimensions, non-flat leaves admit a co-K\"ahler structure, which allows us to realize them as explicit exponential families parametrized by a Lorentz cone. These results establish a concrete bridge between abstract statistical manifolds and exponential families, highlighting deep connections between information geometry, differential geometry, and the topology of 3-manifolds.