Asian Journal of Mathematics

Volume 26 (2022)

Number 6

A hall of statistical mirrors

Pages: 809 – 846



Gabriel Khan (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Jun Zhang (Department of Psychology, University of Michigan, Ann Arbor, Mich., U.S.A.)


The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher–Rao metric and a pair of torsion-free conjugate connections. In recent work [ZK20], the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex-to-symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold.

In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are Kähler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between a Siegel domain and the Siegel–Jacobi space, which are spaces that appear in the context of automorphic forms.


Hessian manifolds, Information geometry, statistical mirror symmetry, Kähler geometry, hyperbolic geometry

2010 Mathematics Subject Classification

26B25, 32Q15, 53C55

This project was supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information Geometry: Geometrization of Science of Information”) and AFOSR Grant FA9550-19-1-0213 (“Brain-Inspired Networks for Multifunctional Intelligent Systems and Aerial Vehicles”), which also supported the first author when he was at the University of Michigan. The first author is currently partially supported by a Simons Collaboration Grant 849022 (“Kähler–Ricci flow and optimal transport”).

Received 28 September 2021

Accepted 13 September 2022

Published 27 April 2023