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

TL;DR

This paper explores the geometric structure of the statistical manifold of normal distributions, revealing its homogeneity and Lie group structure, and characterizes the alpha-connections within this framework.

Contribution

It provides a geometric characterization of alpha-connections on the normal distribution manifold, highlighting its homogeneous Lie group structure.

Findings

Normal distributions form a homogeneous manifold.

The manifold admits a 2D solvable Lie group structure.

Alpha-connections are characterized geometrically on this Lie group.

Abstract

We show that the statistical manifold of normal distributions is homogeneous. In particular, it admits a $2$-dimensional solvable Lie group structure. In addition, we give a geometric characterization of the Amari-Chentsov $\alpha$-connections on the Lie group.