The Fisher-Rao distance is the geodesic distance between probability distributions in a statistical manifold equipped with the Fisher metric, which is a natural choice of Riemannian metric on such manifolds. It has recently been applied to supervised and unsupervised problems in machine learning, in various contexts. Finding closed-form expressions for the Fisher-Rao distance is generally a non-trivial task, and those are only available for a few families of probability distributions. In this survey, we collect examples of closed-form expressions for the Fisher-Rao distance of both discrete and continuous distributions, aiming to present them in a unified and accessible language. In doing so, we also: illustrate the relation between negative multinomial distributions and the hyperbolic model, include a few new examples, and write a few more in the standard form of elliptical distributions.

翻译：Fisher-Rao距离是统计流形上配备Fisher度量（该类流形上黎曼度量的自然选择）的概率分布之间的测地距离。近年来，该距离已被应用于机器学习中的监督与无监督问题，涉及多种场景。寻找Fisher-Rao距离的闭式表达式通常是一项非平凡任务，目前仅对少数概率分布族可行。在本综述中，我们系统整理了离散与连续分布Fisher-Rao距离的闭式表达式实例，旨在以统一且易懂的语言呈现。与此同时，我们还：阐释了负多项分布与双曲模型之间的关联，补充了若干新实例，并将部分分布以椭圆分布的规范形式加以表达。