Uniqueness of the Fisher-Rao metric on the space of smooth densities

TL;DR

This paper proves that on a closed manifold of dimension greater than one, the Fisher-Rao metric is uniquely characterized among diffeomorphism-invariant metrics on the space of smooth densities, up to a scalar multiple.

Contribution

It establishes the uniqueness of the Fisher-Rao metric among all diffeomorphism-invariant weak Riemannian metrics on the space of smooth positive densities.

Findings

Fisher-Rao metric is unique up to scalar multiple on higher-dimensional manifolds.

Any diffeomorphism-invariant weak Riemannian metric on the density space is proportional to Fisher-Rao.

The result extends the understanding of the geometric structure of the space of densities.

Abstract

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.