Bayesian networks are a class of popular graphical models that encode causal and conditional independence relations among variables by directed acyclic graphs (DAGs). We propose a novel structure learning method, annealing on regularized Cholesky score (ARCS), to search over topological sorts, or permutations of nodes, for a high-scoring Bayesian network. Our scoring function is derived from regularizing Gaussian DAG likelihood, and its optimization gives an alternative formulation of the sparse Cholesky factorization problem from a statistical viewpoint. We combine simulated annealing over permutation space with a fast proximal gradient algorithm, operating on triangular matrices of edge coefficients, to compute the score of any permutation. Combined, the two approaches allow us to quickly and effectively search over the space of DAGs without the need to verify the acyclicity constraint or to enumerate possible parent sets given a candidate topological sort. The annealing aspect of the optimization is able to consistently improve the accuracy of DAGs learned by greedy and deterministic search algorithms. In addition, we develop several techniques to facilitate the structure learning, including pre-annealing data-driven tuning parameter selection and post-annealing constraint-based structure refinement. Through extensive numerical comparisons, we show that ARCS outperformed existing methods by a substantial margin, demonstrating its great advantage in structure learning of Bayesian networks from both observational and experimental data. We also establish the consistency of our scoring function in estimating topological sorts and DAG structures in the large-sample limit. Source code of ARCS is available at https://github.com/yeqiaoling/arcs_bn.

中文翻译：

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优化贝叶斯网络基于顺序的学习的正则化 Cholesky 分数


贝叶斯网络是一类流行的图形模型，它通过有向无环图 (DAG) 对变量之间的因果关系和条件独立关系进行编码。我们提出了一种新颖的结构学习方法，即对正则化 Cholesky 分数 (ARCS) 进行退火，以搜索拓扑排序或节点排列，以获得高分贝叶斯网络。我们的评分函数源自正则化高斯 DAG 似然，其优化从统计角度给出了稀疏 Cholesky 分解问题的替代公式。我们将排列空间上的模拟退火与快速近端梯度算法相结合，对边缘系数的三角矩阵进行操作，以计算任何排列的分数。结合起来，这两种方法使我们能够快速有效地搜索 DAG 空间，而无需验证非循环约束或在给定候选拓扑排序的情况下枚举可能的父集。优化的退火方面能够持续提高通过贪婪和确定性搜索算法学习的 DAG 的准确性。此外，我们开发了多种技术来促进结构学习，包括预退火数据驱动的调整参数选择和退火后基于约束的结构细化。通过广泛的数值比较，我们表明 ARCS 大大优于现有方法，证明了它在从观测数据和实验数据中学习贝叶斯网络结构的巨大优势。我们还建立了在大样本限制下估计拓扑排序和 DAG 结构的评分函数的一致性。 ARCS的源代码可以在https://github.com/yeqiaoling/arcs_bn获取。