On Closed-Form Expressions for the Fisher-Rao Distance

Abstract

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.

44 pages, 3 figures, minor corrections. Accepted for publication in Information Geometry

References in corpus (4)

• Adversarial Robustness via Fisher-Rao Regularization
• On Pythagoras' theorem for products of spectral triples
• A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions
• The Fisher-Rao Loss for Learning under Label Noise

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