Manifolds of Differentiable Densities

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 Fréchet 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 $C^{\infty }$.

Version 3: 27 pages. Introduction expanded to discuss applications. Concluding Remarks section added. Improved definition of tangent space (space of signed measures). Discussion of Bayesian data fusion expanded. Discussion of normal charts for the $\alpha$ covariant derivatives added. New references added. No change to results. To appear in ESAIM:Probability and Statistics www.esaim-ps.org