Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold

Uniqueness of the Fisher-Rao metric on the space of smooth densities

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.

Fisher-Rao distance on the covariance cone

The Fisher-Rao geodesic distance on the statistical manifold consisting of zero-mean p-dimensional multivariate Gaussians appears without proof in several places (such as Steven Smith's "Covariance, Subspace, and Intrinsic Cramer-Rao Bounds"). In this paper, we give a proof using basic Riemannian geometry.

Fisher-Rao geometry of Dirichlet distributions

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fr{é}chet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

无限维希尔伯特空间上等价高斯测度的Fisher-Rao几何

Fisher-Rao geometry of equivalent Gaussian measures on infinite-dimensional Hilbert spaces

本文在无限维希尔伯特空间上显式描述了等价中心高斯测度的Fisher-Rao黎曼度量，证明有限维高斯密度相关几何量可自然推广，并建立了与正定单位化希尔伯特-施密特算子黎曼几何的联系。

The Fisher-Rao geometry of beta distributions applied to the study of canonical moments

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrinsic representation of the moments of a probability distribution. Drawing on the fact that a uniform distribution in the regular moment space corresponds to a product of beta distributions in the canonical moment space, we propose a mapping from the space of canonical moments to t…

确定性测度传输的平流Fisher-Rao几何

The Advective Fisher-Rao Geometry of Deterministic Measure Transport

本文提出平流Fisher-Rao度量，从三视角推导其来源，经实验验证该度量可最优拟合概率密度，高斯-牛顿法可最优拟合速度场，为概率测度路径优化提供新几何工具。