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

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.