Version 1
: Received: 14 March 2018 / Approved: 14 March 2018 / Online: 14 March 2018 (14:08:11 CET)
How to cite:
Fong, S.; Tino, P. Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds. Preprints2018, 2018030106. https://doi.org/10.20944/preprints201803.0106.v1
Fong, S.; Tino, P. Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds. Preprints 2018, 2018030106. https://doi.org/10.20944/preprints201803.0106.v1
Fong, S.; Tino, P. Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds. Preprints2018, 2018030106. https://doi.org/10.20944/preprints201803.0106.v1
APA Style
Fong, S., & Tino, P. (2018). Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds. Preprints. https://doi.org/10.20944/preprints201803.0106.v1
Chicago/Turabian Style
Fong, S. and Peter Tino. 2018 "Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds" Preprints. https://doi.org/10.20944/preprints201803.0106.v1
Abstract
This paper aims to describe the geometrical structure and explicit expressions of family of finitely parametrized probability densities over smooth manifold $M$. The geometry of family of probability densities on $M$ are inherited from probability densities on Euclidean spaces $\left\{U_\alpha \right\}$ via bundle morphisms, induced by an orientation-preserving diffeomorphisms $\rho_\alpha:U_\alpha \rightarrow M$. Current literature inherits densities on $M$ from tangent spaces via Riemannian exponential map $\exp: T_x M \rightarrow M$; densities on $M$ are defined locally on region where the exponential map is a diffeomorphism. We generalize this approach with an arbitrary orientation-preserving bundle morphism; we show that the dualistic geometry of family of densities on $U_\alpha$ can be inherited to family of densities on $M$. Furthermore, we provide explicit expressions for parametrized probability densities on $\rho_\alpha(U_\alpha) \subset M$. Finally, using the component densities on $\rho_\alpha(U_\alpha)$, we construct parametrized mixture densities on totally bounded subsets of $M$. We provide a description of inherited mixture product dualistic geometry of the family of mixture densities.
Keywords
Probability densities on manifold, geometric statistics, Hessian manifold
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.