A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions

Abstract

We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. 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)×(d+1) symmetric positive–definite matrices. 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 Calvo and Oller’s isometric embedding.

Keywords

• Fisher–Rao normal manifold
• symmetric positive–definite matrix cone
• isometric embedding
• information geometry
• exponential family
• elliptical distribution