Preprint Article Version 1 This version is not peer-reviewed

# Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds

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. Preprints 2018, 2018030106 (doi: 10.20944/preprints201803.0106.v1). Fong, S.; Tino, P. Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds. Preprints 2018, 2018030106 (doi: 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.

## Subject Areas

Probability densities on manifold, geometric statistics, Hessian manifold

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