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

Understand

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.