  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, 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 K\"ahler
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 the Siegel half-space and the
Siegel-Jacobi space, which are spaces that appear in the context of automorphic
forms.
