Smoothness of Gaussian conditional independence models

TL;DR

This paper investigates the geometric structure of Gaussian conditional independence models, revealing that some models, though representable, can decompose into unions of smooth submodels, impacting statistical inference.

Contribution

It provides the first detailed analysis of the smoothness and decomposition properties of Gaussian conditional independence models involving up to four variables.

Findings

Some Gaussian CI models decompose into unions of smooth submodels.

Certain probabilistically representable CI relations lead to non-trivially decomposable models.

The study enhances understanding of the geometric complexity in Gaussian CI models.

Abstract

Conditional independence in a multivariate normal (or Gaussian) distribution is characterized by the vanishing of subdeterminants of the distribution's covariance matrix. Gaussian conditional independence models thus correspond to algebraic subsets of the cone of positive definite matrices. For statistical inference in such models it is important to know whether or not the model contains singularities. We study this issue in models involving up to four random variables. In particular, we give examples of conditional independence relations which, despite being probabilistically representable, yield models that non-trivially decompose into a finite union of several smooth submodels.