Geometric mean of probability measures and geodesics of Fisher information metric

TL;DR

This paper explores the geometric structure of the space of probability measures on a manifold using the Fisher information metric, introducing a geometric mean and characterizing geodesics and distances.

Contribution

It defines a geometric mean for probability measures and characterizes geodesics and distances in the Fisher information metric space, establishing uniqueness and minimality of geodesics.

Findings

Geodesics are expressed via the normalized geometric mean.

Any two probability measures are connected by a unique geodesic.

The function efined by the arccosine of the integral of square roots gives the Riemannian distance.

Abstract

The space of all probability measures having positive density function on a connected compact smooth manifold $M$, denoted by $\mathcal{P}(M)$, carries the Fisher information metric $G$. We define the geometric mean of probability measures by the aid of which we investigate information geometry of $\mathcal{P}(M)$, equipped with $G$. We show that a geodesic segment joining arbitrary probability measures $\mu_1$ and $\mu_2$ is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of $\mathcal{P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function $\ell$ defined by $\ell(\mu_1, \mu_2):=2\arccos\int_M \sqrt{p_1\,p_2}\,d\lambda$, $\mu_i=p_i\,\lambda$, $i=1,2$ gives the Riemannian distance function on $\mathcal{P}(M)$. It is shown that geodesics are all minimal.