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Conformal Holonomy of the Bivariate Gaussian Manifold

James A. Reid

TL;DR

This work determines the conformal holonomy of the Fisher-Rao metric on the bivariate Gaussian information-manifold. Using tractor calculus and conformal holonomy theory, it proves that the full bivariate Gaussian manifold has irreducible conformal holonomy $ ext{$Hol(\mathcal{G},[g]) = SO^{0}(1,6)$}$, while the independence submanifold (the product of univariate Gaussians) exhibits a reducible holonomy and reduces to $ ext{$Hol(\mathcal{I},[g]) = SO^{0}(1,4)$}$ due to the presence of an Einstein metric in its conformal class with $\mathsf{scal}=-2$. Key steps include ruling out degenerate and non-degenerate holonomy-invariant subspaces, establishing simple connectivity, and invoking conformal holonomy classifications tied to parallel tractors and Einstein metrics. The results reveal how conformal holonomy distinguishes generic from special geometric structures in statistical manifolds and connects information geometry with deep conformal-geometric invariants.

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 $SO^{0}(1,4)$.

Conformal Holonomy of the Bivariate Gaussian Manifold

TL;DR

This work determines the conformal holonomy of the Fisher-Rao metric on the bivariate Gaussian information-manifold. Using tractor calculus and conformal holonomy theory, it proves that the full bivariate Gaussian manifold has irreducible conformal holonomy Hol(\mathcal{G},[g]) = SO^{0}(1,6), while the independence submanifold (the product of univariate Gaussians) exhibits a reducible holonomy and reduces to Hol(\mathcal{I},[g]) = SO^{0}(1,4) due to the presence of an Einstein metric in its conformal class with . Key steps include ruling out degenerate and non-degenerate holonomy-invariant subspaces, establishing simple connectivity, and invoking conformal holonomy classifications tied to parallel tractors and Einstein metrics. The results reveal how conformal holonomy distinguishes generic from special geometric structures in statistical manifolds and connects information geometry with deep conformal-geometric invariants.

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, . Remarkably, however, when the random variables are statistically independent, the conformal holonomy representation is reducible and the conformal holonomy group is .
Paper Structure (8 sections, 19 theorems, 64 equations, 1 table)

This paper contains 8 sections, 19 theorems, 64 equations, 1 table.

Table of Contents

  1. Information geometry
  2. Preliminaries
  3. Conformal geometry
  4. Conformal structures and tractor bundles
  5. Conformal holonomy theory
  6. Conformal holonomy of the bivariate Gaussian manifold
  7. Conformal holonomy of the bivariate Gaussian manifold
  8. Conformal holonomy of the independence submanifold

Key Result

Theorem 1

Let $(M,c)$ be a conformal manifold of dimension $\geq 3$. If the conformal class $c$ contains an Einstein metric $g$, then there exists a non-trivial section $V \in \Gamma(\mathcal{T})$ with $\nabla^{\mathcal{T}} V =0$. On the other hand, if we suppose that there exists a nontrivial $\nabla^{\mathc $\square$

Theorems & Definitions (28)

  • Theorem 1: baumjuhl
  • Theorem 2: baumreview
  • Theorem 3: Berger baumreview
  • Corollary 1: baumreview
  • Corollary 2: galaevleistner
  • Theorem 4: baumjuhl
  • Theorem 5: alt
  • Theorem 6: baumjuhl
  • Theorem 7: baumjuhl
  • Proposition 1
  • ...and 18 more