Construction of Exponential Families from Statistical Manifolds

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-Kähler 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.

💡 Research Summary

The paper tackles a central open problem in information geometry: how to realize a given statistical manifold as an exponential family. Building on the classical Fisher information metric, Amari–Čencov α‑connections, and the notion of statistical manifolds introduced by Lauritzen, the author proves that every compact statistical manifold admits a singular foliation whose leaves are Hessian manifolds. The construction proceeds by solving the second covariant derivative equation (E(\nabla):\nabla^{2}X=0) for vector fields, which yields a finite‑dimensional associative algebra (\mathcal J^{\nabla}). By Lie’s third theorem this algebra integrates to a simply‑connected Lie group (G^{\nabla}) acting on the manifold; the orbit decomposition defines the foliation (F^{\nabla}). Each leaf is (\nabla)-autoparallel, and the restriction of (\nabla) to a leaf together with the induced metric is shown to be dually flat, i.e., a Hessian manifold.

The topology of compact orientable leaves is then analyzed via the first Koszul form (\beta). If (\beta=0), the Levi‑Civita connection coincides with the flat affine connection, and the leaf is a flat Riemannian manifold. Classical Bieberbach and Wolf theorems imply that such a leaf is finitely covered by a torus. If (\beta\neq0), (\beta) is parallel with respect to the Levi‑Civita connection, its norm is constant, and the leaf acquires a co‑Kähler structure. In three dimensions, a non‑flat co‑Kähler leaf necessarily has all odd Betti numbers, a fact that aligns with known constraints on co‑Kähler manifolds.

The most striking contribution is the explicit realization of a three‑dimensional non‑flat leaf as a quotient of an exponential family. The author exhibits a Lorentz cone ({x\in\mathbb R^{1,n}\mid \langle x,x\rangle>0}) as the natural parameter space, defines a family \