Learning the graphical structure of Bayesian networks is key to describing data-generating mechanisms in many complex applications but poses considerable computational challenges. Observational data can only identify the equivalence class of the directed acyclic graph underlying a Bayesian network model, and a variety of methods exist to tackle the problem. Under certain assumptions, the popular PC algorithm can consistently recover the correct equivalence class by reverse-engineering the conditional independence (CI) relationships holding in the variable distribution. The dual PC algorithm is a novel scheme to carry out the CI tests within the PC algorithm by leveraging the inverse relationship between covariance and precision matrices. By exploiting block matrix inversions we can also perform tests on partial correlations of complementary (or dual) conditioning sets. The multiple CI tests of the dual PC algorithm proceed by first considering marginal and full-order CI relationships and progressively moving to central-order ones. Simulation studies show that the dual PC algorithm outperforms the classic PC algorithm both in terms of run time and in recovering the underlying network structure, even in the presence of deviations from Gaussianity. Additionally, we show that the dual PC algorithm applies for Gaussian copula models, and demonstrate its performance in that setting.

翻译：学习贝叶斯网络的图形结构是描述许多复杂应用中数据生成机制的关键，但面临着巨大的计算挑战。观测数据只能识别贝叶斯网络模型下有向无环图的等价类，已有多种方法来解决这一问题。在特定假设下，流行的PC算法可以通过逆向分析变量分布中成立的条件独立性关系，一致地恢复正确的等价类。双PC算法是一种新颖的方案，它利用协方差矩阵与精度矩阵之间的逆关系，在PC算法内部执行条件独立性检验。通过利用分块矩阵求逆，我们还可以对互补（或对偶）条件集的偏相关系数进行检验。双PC算法的多重条件独立性检验首先考虑边际和全阶条件独立性关系，然后逐步过渡到中心阶关系。模拟研究表明，即使在高斯性偏离的情况下，双PC算法在运行时间和恢复底层网络结构方面均优于经典PC算法。此外，我们证明双PC算法适用于高斯copula模型，并展示了其在该场景下的性能。