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

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