## JMI2011B-6 Remarks on geodesics for multivariate normal models (pp.125-130)

Author(s)： Takuro Imai, Akira Takaesu and Masato Wakayama

J. Math-for-Ind. **3B** (2011) 125-130.

- File： JMI2011B-6.pdf (130KB)

Abstract

A complete description of the geodesic curves on the Riemann manifold of multivariate normal distributions equipped with the Fisher information metric has been accomplished by Eriksen in 1987, and later by Calvo and Oller in 1991 but in a different manner. The former describes geodesic curves in terms of an exponential map in somewhat mysterious way and the latter obtains a solution of the differential equation of a geodesic curve explicitly by solving much general system of differential equations. The method what Erikson had taken seems to have a group theoretinature while it is still unclear. The purposes of this short note are to derive the explicit formula of the geodesic curve from the result obtained by Eriksen and to clarify why such exponential map may give geodesic curves for the one dimensional normal distribution case.

Keyword(s). geodesics, multivariate normal distribution, statistical manifold, Fisher's informationmetric, Riemann symmetric spaces, Lorentz group.