Algebraic geometry of Gaussian Bayesian networks

TL;DR

This paper explores the algebraic structure of Gaussian Bayesian networks, revealing how their models relate to classical algebraic varieties and extending the recursive factorization theorem to hidden variable scenarios.

Contribution

It demonstrates that Bayesian network ideals are generated by graph-implied conditional independences and are homogeneous under a specific multigrading, linking these models to classical algebraic geometry.

Findings

For tree-structured graphs, the ideal is generated by graph-implied CI statements.

Bayesian network ideals are homogeneous under a multigrading.

Connections are established between Bayesian network ideals and classical algebraic varieties.

Abstract

Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem to situations with hidden variables. In the case when the underlying graph is a tree, we show that the vanishing ideal of the model is generated by the conditional independence statements implied by graph. We also show that the ideal of any Bayesian network is homogeneous with respect to a multigrading induced by a collection of upstream random variables. This has a number of important consequences for hidden variable models. Finally, we relate the ideals of Bayesian networks to a number of classical constructions in algebraic geometry including toric degenerations of the Grassmannian, matrix Schubert varieties, and secant varieties.