A characterization of the alpha-connections on the statistical manifold of multivariate normal distributions

Abstract

We study a statistical manifold $(\mathcal{N},g^F,{\nabla }^A,{\nabla }^{A\ast })$ of multivariate normal distributions, where $g^F$ is the Fisher metric and ${\nabla }^A$ is the Amari-Chentsov connection and ${\nabla }^{A\ast }$ is its conjugate connection. We will show that it admits a solvable Lie group structure and moreover the Amari-Chentsov connection ${\nabla }^A$ on $(\mathcal{N},g^F)$ will be characterized by the conjugate symmetry, i.e., a curvatures identity $R=R^{\ast }$ of a connection $\nabla$ and its conjugate connection ${\nabla }^{\ast }$.