Construction of Exponential Families from Statistical Manifolds
Emmanuel Gnandi
TL;DR
The paper tackles the problem of which statistical manifolds can be realized as exponential families by proving that every compact statistical manifold admits a singular foliation with Hessian leaves. Compact orientable leaves are shown to be either finite quotients of flat tori or mapping tori with periodic monodromy, while 3D non-flat leaves are quotients of exponential families and have odd Betti numbers. The authors construct a finite-dimensional Lie algebra of ∇-parallel vector fields solving ∇^2X=0, integrate to a foliation by Lie-group orbits, and use the first Koszul form to constrain leaf topology, culminating in an explicit Koszul–Vinberg realization on the Lorentz cone Θ ≅ H^2×R. This connection between Hessian geometry and exponential families advances understanding of which manifolds can arise from exponential-family models and provides concrete geometric constructions for realizing them.
Abstract
This paper addresses the fundamental question of constructing exponential families from statistical manifolds. We show that every compact statistical manifold admits a singular foliation whose leaves are Hessian manifolds. Moreover, any non-flat, compact, orientable 3-dimensional leaf appears as a quotient of an exponential family and has only odd Betti.