Any K\"ahler metric is a Fisher information metric

TL;DR

This paper demonstrates that all K"ahler metrics can be viewed as Fisher information metrics, establishing a statistical interpretation of K"ahler and co-K"ahler manifolds and linking geometric structures with information theory.

Contribution

It provides a novel characterization of K"ahler and co-K"ahler manifolds as Fisher information metrics, connecting complex geometry with statistical models.

Findings

K"ahler and co-K"ahler manifolds can be seen as parametric probability families.

Every real analytic K"ahler metric is locally a Fisher information metric.

Links between K"ahler potential and Kullback-Leibler divergence are established.

Abstract

The Fisher information metric or the Fisher-Rao metric corresponds to a natural Riemannian metric defined on a parameterized family of probability density functions. As in the case of Riemannian geometry, we can define a distance in terms of the Fisher information metric, called the Fisher-Rao distance. The Fisher information metric has a wide range of applications in estimation and information theories. Indeed, it provides the most informative Cramer-Rao bound for an unbiased estimator. The Goldberg conjecture is a well-known unsolved problem which states that any compact Einstein almost K\"ahler manifold is necessarily a K\"ahler-Einstein. Note that, there is also a known odd-dimensional analog of the Goldberg conjecture in the literature. The main objective of this paper is to establish a new characterization of coK\"ahler manifolds and K\"ahler manifolds; our characterization is statistical in nature. Finally, we corroborate that every, K\"ahler and co-K\"ahler manifolds, can be viewed as being a parametric family of probability density functions, whereas K\"ahler and coK\"ahler metrics can be regarded as Fisher information metrics. In particular, we prove that, when the K\"ahler metric is real analytic, it is always locally the Fisher information of an exponential family. We also tackle the link between K\"ahler potential and Kullback-Leibler divergence.