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

TL;DR

This paper characterizes the alpha-connections on the statistical manifold of multivariate normal distributions, revealing a Lie group structure and a curvature identity that uniquely defines the Amari-Chentsov connection.

Contribution

It demonstrates that the manifold admits a solvable Lie group structure and characterizes the Amari-Chentsov connection via conjugate symmetry and curvature identities.

Findings

The manifold has a solvable Lie group structure.

The Amari-Chentsov connection is characterized by conjugate symmetry.

Curvature identities uniquely determine the connection.

Abstract

We study a statistical manifold $(\mathcal{N}, g^F, \nabla^{A}, \nabla^{A*})$ of multivariate normal distributions, where $g^F$ is the Fisher metric and $\nabla^{A}$ is the Amari-Chentsov connection and $\nabla^{A*}$ 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 $R=R^*$ of a connection $\nabla$ and its conjugate connection $\nabla^*$.