Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks

TL;DR

This paper introduces a novel algebraic approach using Gr"obner bases to estimate the marginal independence structure of Bayesian networks, improving accuracy over traditional independence tests.

Contribution

It develops an algebraic framework connecting Bayesian network structures to graph invariants and proposes a new MCMC method, GrUES, for structure estimation.

Findings

GrUES outperforms simple independence tests in recovering true structures.

The algebraic characterization links independence graphs to intersection and independence numbers.

High coverage of true structures in credible sets for sufficiently dense data-generating graphs.

Abstract

We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data, learning an undirected graph we call the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr\"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr\"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the $20\%$ HPD credible sets include the true structure at a high rate for data-generating graphs with density at least $0.5$.