# midpoint between two normal distributions for the Rao-Fisher metric

Given two multivariate gaussian distributions $G_0 \sim N(\mu_0,\Omega_0)$ and $G_1 \sim N(\mu_1,\Omega_1)$, is there a closed-form formula for the gaussian distribution equidistant from them that is located on the geodesic for the Rao-Fisher metric? If not, are there other metrics with a statistical meaning where it is available?

Yes, there is, because the the Fisher-Rao metric on the space of multivariate Gaussian distributions is isometric to a metric on the symmetric space $SL(n+1)/SO(n+1)$ which is a close relative to its metric as a symmetric space of non-compact type. See the following paper. You have to adapt the symmetric space formulas for geodesics to this slightly different metric.

• MR1790612 Lovrić, Miroslav; Min-Oo, Maung; Ruh, Ernst A.: Multivariate normal distributions parametrized as a Riemannian symmetric space. J. Multivariate Anal. 74 (2000), no. 1, 36–48.

It is also known that the Fisher Rao metric on the space of all densities ($L^2$ or smooth) is isometric to a an open subset of sphere in a pre-Hilbert space. See (here).

• The Fisher metric for multivariate Gaussian distributions is not symmetric, and doesn't have a closed form midpoint formula. For details, see the paper "A Riemannian Geometry of the Multivariate Normal Model" by Skovgaard. The model introduced in your reference is not the Fisher metric, but a slightly different metric where you can calculate the midpoint. Commented Oct 6, 2020 at 21:48

There is no closed-form formula. We can compute the Euler-Poincaré equation and with symmetries we can reduce this equation to Euler-Poincaré equation. We can then use "geodesic shooting" to compute the distance between the 2 multivariate gaussian distributions. See the following reference: [A] M. Pilté and F. Barbaresco, "Tracking quality monitoring based on information geometry and geodesic shooting," 2016 17th International Radar Symposium (IRS), Krakow, 2016, pp. 1-6. available on IEEExplore: http://ieeexplore.ieee.org/document/7497346/ To understand the geometric meaning of the Euler-Poincaré equation of this geodesic could be understand by use of "moment map" from Jean-Marie Souriau that is an element of Lie algebra (when you have symmetries, the moment map is invariant; its components are the Emmy Noether invariant; but Souriau Moment map has a geometric meaning). You can read more details on this geometric approach in 2 papers: [B] Barbaresco, F. Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families. Entropy 2016, 18, 386. http://www.mdpi.com/1099-4300/18/11/386

All these tools will be debated in GSI'17 conference and CIRM TGSI'17 seminar.

F. Barbaresco GSI'17 Co-chairman