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?
 A: 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.


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*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).
A: 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
