Newton, NJ (2018) Manifolds of Differentiable Densities. ESAIM: Probability and Statistics, 22. pp. 19-34. DOI https://doi.org/10.1051/ps/2018003
Newton, NJ (2018) Manifolds of Differentiable Densities. ESAIM: Probability and Statistics, 22. pp. 19-34. DOI https://doi.org/10.1051/ps/2018003
Newton, NJ (2018) Manifolds of Differentiable Densities. ESAIM: Probability and Statistics, 22. pp. 19-34. DOI https://doi.org/10.1051/ps/2018003
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 $C^\infty$.
Item Type: | Article |
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Additional Information: | 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 |
Uncontrolled Keywords: | Fisher-Rao Metric; Banach Manifold; Frechet Manifold; Information Geometry; Nonparametric Statistics |
Subjects: | Q Science > QA Mathematics |
Divisions: | Faculty of Science and Health Faculty of Science and Health > Computer Science and Electronic Engineering, School of |
SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |
Depositing User: | Unnamed user with email elements@essex.ac.uk |
Date Deposited: | 26 Jun 2018 08:32 |
Last Modified: | 16 May 2024 19:13 |
URI: | http://repository.essex.ac.uk/id/eprint/22320 |
Available files
Filename: mosdgen_ar3.pdf