# Information geometry and asymptotic geodesics on the space of normal distributions

## TL;DR

This paper explores the geometric structure of the space of multivariate normal distributions, comparing Fisher and Killing metrics, and analyzes when the Killing metric effectively approximates Fisher geodesics for long distances.

## Contribution

It provides a detailed survey of the geometry of the normal distribution space and quantifies the approximation quality of Killing geodesics to Fisher geodesics.

## Key findings

- Killing metric approximates Fisher geodesics well for long distances.
- The space of normal distributions is a Riemannian manifold with complex geometric properties.
- Quantitative analysis of geodesic defects supports the use of Killing metric in computations.

## Abstract

The family $\mathcal{N}$ of $n$-variate normal distributions is parameterized by the cone of positive definite symmetric $n\times n$-matrices and the $n$-dimensional real vector space. Equipped with the Fisher information metric, $\mathcal{N}$ becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space $Pos_1(n+1,\mathbb{R})$ of unimodular positive definite symmetric $(n+1)\times(n+1)$-matrices. As the computation of distances in the Fisher metric for $n>1$ presents some difficulties, Lovri\v{c} et al.~(2000) proposed to use the Killing metric on $Pos_1(n+1,\mathbb{R})$ as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space $\mathcal{N}$ and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances.