A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions

TL;DR

This paper introduces a simple numerical method to approximate the Fisher-Rao distance between multivariate normal distributions using discretized curves and divergence measures, validated through experiments and geometric analysis.

Contribution

The paper proposes a novel approximation technique for Fisher-Rao distance based on discretizing curves and divergence measures, and compares different parameterizations and embeddings.

Findings

The approximation method closely matches bounds for Fisher-Rao distance.

Linear and geometric curves yield different approximation qualities.

The geometric properties of the embedding are characterized.

Abstract

We present a simple method to approximate Rao's distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating Rao's distances between successive nearby normal distributions on the curves by the square root of Jeffreys divergence, the symmetrized Kullback-Leibler divergence. We consider experimentally the linear interpolation curves in the ordinary, natural and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller's isometric embedding of the Fisher-Rao $d$-variate normal manifold into the cone of $(d+1)\times (d+1)$ symmetric positive-definite matrices [Journal of multivariate analysis 35.2 (1990): 223-242]. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information-geometric properties of the Calvo and Oller's isometric embedding.