Manifolds of Differentiable Densities

TL;DR

This paper constructs infinite-dimensional manifolds of differentiable probability densities with rich geometric structures, including Fisher-Rao metric and Amari's covariant derivatives, extending non-parametric statistical geometry.

Contribution

It introduces a family of infinite-dimensional manifolds of differentiable densities with comprehensive geometric structures, including the Fisher-Rao metric and $orall \, ext{α}$-covariant derivatives.

Findings

Manifolds are $C^ $-embedded in finite measure manifolds.

They are dually flat with mixture and exponential charts.

Curvatures with respect to $ ext{α}$-covariant derivatives are derived.

Abstract

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fr\'{e}chet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's $\alpha$-covariant derivatives for all $\alpha\in R$. By construction, they are $C^\infty$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually ($\alpha=\pm 1$) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $\alpha$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $\alpha$-divergences are of class $C^\infty$.