The question, as posed, doesn't make sense. The term ``geodesic distance'' makes sense once a metric has been defined, that defines the ``distance'' between ``points'' (here these would be probability distributions) in this space. And from this definition one can define the  length of ``parametric curves'' on this space and, thus, arrive at the definition of geodesics.  

One way to carry out this procedure was found by Rao: http://www.scholarpedia.org/article/Fisher-Rao_metric

Hi,

Thank you for pointing this out. I was implicitly assuming that the distance between two infinitesimally was the Fisher-Rao metric that you mentioned in that link.

In the literature I have found the case of two multivariate Gaussians, but only in particular cases, e.g. same covariance matrix or same expected value (see e.g.  S.I.Amari et al, Differential geometry in statistical inference, IMS Lecture Notes Monogr. Ser.

Volume 10, 1987 - Chapter 5,.6).

My best regards, Giuseppe Papari

Once the metric is known, one has to solve the geodesic equation for the ``parametric curve'' in this space. For some explicit examples, cf. Rao's paper: http://projecteuclid.org/euclid.lnms/1215467062

The idea is the clear: there's a family of distributions, f(x,a_k), labeled by the parameters, a_k, k=1,…D. For the case at hand, these would be the mean and variance parameters of the Gaussian distributions. One obtains the metric, from that the affine connection and one solves the geodesic equation, with the affine parameter, such that for the value, say, λ=0 one obtains one distribution, for the value, say, λ=1, one obtains the other. The length of the parametric curve, with respect to this metric,  thus obtained, is the distance. Of course it may be necessary to solve the equation numerically. 

Hi Nicolis,

in the paper you mention (the same I referred in my previous question) they only give explicit formulas for the cases in which the two Gaussians have either the same mean, or the same covariance matrix.

In general, you may need to solve integrals numerically. I was wandering if anyone knows an analytical expression for the general Gaussian case.

No, Rao's paper, which I link to,  isn't the same as Amari's paper. It's a good idea to actually read it. 

Hi,

The Amari source I mentioned is not a paper, it is a collection of papers. The fifth chapter is indeed authored By rao and it was the same you linked and it only contains te formulas for the "same mean" and "same covariance" cases.

Anyway... Do you know a closed form for the General Gaussian case?

It's more useful to learn how the known expressions have been found so as to apply the method to other cases and develop, among other things, the code to do so. That's what matters. ``Closed forms'' aren't that interesting as such-it's an illusion to believe that they, somehow, contain more information than a  calculational method, that can be shown to provide reliable results. The time spent in searching for formulas is better spent in learning how to calculate them. I don't know a closed form, beyond what's provided by Rao, but I think that it's not that relevant. Knowing how to solve the geodesic equation and how to use its solution is relevant.

This is a year-old question so apologies if it's a faux pas to reply here. I do not believe there is a closed form expression for the geodesic curve connecting two arbitrary multivariate normals. There are closed-form expressions for a geodesic curve on this manifold, given some initial starting direction (see Imai 2011 in the Journal of Maths-for-Industry for the easier-to-understand explanation of this) but I do not believe anyone has figured out how to analytically reverse-engineer the correct starting direction to arrive at a desired target point, nor a way to express the integral of infinitesimal lengths along this curve in some closed-form expression.