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.

Figures (3)

• Figure 1: Schematic representing the parametrisation $\varphi$ from the parameter space $\Xi$ to the statistical manifold $\mathcal{S}$. The curve $\gamma(t)$ joins two points in the manifold, which are probability density (or mass) functions.
• Figure 2: Geodesics joining points $p=(0.7,0.2,0.1)$ and $q=(0.1,0.3,0.6)$ according to categorical (solid) and negative multinomial (dashed) metrics, seen in the parameter space, on the simplex, and on the sphere. The distance between the categorical distributions is $d_{\text{cat}}(p,q) \approx 1.432$, and between the negative multinomial distributions is $d_{\text{neg-mult}}(p,q) \approx 2.637 \sqrt{x_n}$.
• Figure 3: Geodesics joining points $(\mu_1,\sigma_1)=(2,0.5)$ and $(\mu_2,\sigma_2)=(5,1)$ according to Gaussian metric (solid) and Cauchy metric (dashed), seen in the parameter space $(\mu,\sigma)$, and the corresponding densities. The distance between the two Gaussian distributions is $d_{\text{Gaussian}}((\mu_1,\sigma_1),(\mu_2,\sigma_2)) \approx 3.443$, and between the Cauchy distributions is $d_{\text{Cauchy}}((\mu_1,\sigma_1),(\mu_2,\sigma_2)) \approx 1.721$.

Theorems & Definitions (21)

• Proposition 1.1: calin2014
• proof
• Proposition 1.2: calin2014
• proof
• Proposition 1.3: calin2014
• proof
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• proof