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 $0.5$.

翻译：我们考虑从观测数据中估计贝叶斯网络边际独立结构的问题，即学习一种称为无条件依赖图的无向图。研究表明，贝叶斯网络的无条件依赖图对应于具有相同独立数和相交数的图。基于这一观察，我们给出了与贝叶斯网络无条件依赖图相关的环面理想的一组Gröbner基，并通过额外的二项式关系将其扩展，以连接所有此类图的空间。基于由此生成的移动，实现了一种名为GrUES（基于Gröbner的无条件等价搜索）的MCMC方法，并将其应用于合成高斯数据。GrUES通过惩罚最大似然或最大后验估计，以比简单独立性检验更高的比率恢复真实的边际独立结构，同时还能给出后验估计值，其中对于密度至少为0.5的数据生成图，20%的最高后验密度可信集以高比率包含真实结构。