论文标题
关于贝叶斯网络边际独立结构的组合和代数观点
Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks
论文作者
论文摘要
我们考虑从观察数据估算贝叶斯网络边际独立性结构的问题，学习一个无方向的图表，我们称为无条件依赖图。我们表明，贝叶斯网络的无条件依赖图对应于具有相等独立性和相交数量的图形。使用此观察结果，给出了与贝叶斯网络无条件依赖图相关的曲折理想的基础，然后通过其他二项式关系扩展，以连接所有此类图的空间。 MCMC方法，称为Grues（基于Gröbner的无条件等效搜索），是根据结果移动实现的，并应用于合成高斯数据。 GRUE通过受惩罚的最大似然或MAP估算的速度恢复了真正的边际独立性结构，速度比简单的独立性测试较高，同时还产生了后部的估计，其中20美元\％$ $ HPD可信集包括以高度的数据生成图的较高速率，并至少$ 0.5 $。
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 Gröbner 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 (Gröbner-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 $0.5$.
