When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher-Rao information geometry. Interestingly, this yields a point of view that allows for leveragingmany tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cram\'er-Rao bounds, and classification using Riemannian distances.

翻译：在处理参数化统计模型时，通过为参数空间赋予Fisher信息度量，可以自然地生成黎曼流形。该度量在参数上诱导的几何称为Fisher-Rao信息几何。有趣的是，这种视角使我们能够利用微分几何中的多种工具。在简要介绍这些概念后，我们将展示这些几何工具在椭圆分布框架中的一些实际应用。本报告的第二部分分为三个主要方向：协方差矩阵估计的黎曼优化、内在克拉美-罗界以及基于黎曼距离的分类。