Volume 22, 2018
|Page(s)||19 - 34|
|Published online||04 October 2018|
Manifolds of differentiable densities
School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park,
CO4 3SQ, UK.
* Corresponding author: firstname.lastname@example.org
Accepted: 5 January 2018
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 Cbk with respect to appropriate reference measures. The case k = ∞, 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 α-covariant derivatives for all α ∈ ℝ. By construction, they are C∞-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C∞.
Mathematics Subject Classification: 46A20 / 60D05 / 62B10 / 62G05 / 94A17
Key words: Fisher-Rao Metric / Banach manifold / Fréchet manifold / information geometry / non-parametric statistics.
© EDP Sciences, SMAI 2018
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