Symplectic and Kähler structures on statistical manifolds induced from divergence functions

Abstract

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, non-negative function on the product manifold M x M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold Mx. It is well-known (Eguchi, 1982) that the statistical structure on M (a Riemmanian metric with a pair of conjugate affine connections) can be constructed from the second and third derivatives of D evaluated at Mx. Here, we investigate Riemannian and symplectic structures on M x M as induced from D. We derive a necessary condition about D for M x M to admit a complex representation and thus become a Kähler manifold. In particular, Kähler potential is shown to be globally defined for the class of Φ-divergence induced by a strictly convex function Φ (Zhang, 2004). In such case, we recov er α-Hessian structure on the diagonal manifold M x, which is equiaffine and displays the so-called "reference-representation biduality". © 2013 Springer-Verlag.