Conformal Holonomy of the Bivariate Gaussian Manifold

TL;DR

This paper investigates the conformal geometry of statistical manifolds, specifically the bivariate Gaussian manifold, revealing how holonomy groups change with independence of variables, thus opening new avenues in information geometry.

Contribution

It determines the conformal holonomy groups of the Fisher-Rao metric on the bivariate Gaussian manifold, highlighting differences between dependent and independent variables.

Findings

Holonomy group is $SO^{0}(1,6)$ for generic variables.

Holonomy reduces to $SO^{0}(1,4)$ when variables are independent.

Conformal invariants of the Fisher-Rao metric are characterized.

Abstract

Statistical manifolds, the parameter spaces of smooth families of probability density functions, are the central objects of study in information geometry. While the elementary differential geometry of Riemannian statistical manifolds is well-known, their conformal geometry remains entirely unexplored. In this article, we begin this programme of exploration by determining some invariants of the conformal structure of the Fisher-Rao metric. Specifically, we study the holonomy of a conformally-invariant connection on the standard tractor bundle of the bivariate Gaussian manifold. It is found that for a generic pair of random variables, the conformal holonomy group is the identity-connected component of the indefinite special orthogonal group, $SO^{0}(1,6)$. Remarkably, however, when the random variables are statistically independent, the conformal holonomy representation is reducible and the conformal holonomy group is $SO^{0}(1,4)$.