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: email@example.com
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.