Score-based methods for learning Bayesain networks(BN) aim to maximizing the global score functions. However, if local variables have direct and indirect dependence simultaneously, the global optimization on score functions misses edges between variables with indirect dependent relationship, of which scores are smaller than those with direct dependent relationship. In this paper, we present an identifiability condition based on a determined subset of parents to identify the underlying DAG. By the identifiability condition, we develop a two-phase algorithm namely optimal-tuning (OT) algorithm to locally amend the global optimization. In the optimal phase, an optimization problem based on first-order Hilbert-Schmidt independence criterion (HSIC) gives an estimated skeleton as the initial determined parents subset. In the tuning phase, the skeleton is locally tuned by deletion, addition and DAG-formalization strategies using the theoretically proved incremental properties of high-order HSIC. Numerical experiments for different synthetic datasets and real-world datasets show that the OT algorithm outperforms existing methods. Especially in Sigmoid Mix model with the size of the graph being ${\rm\bf d=40}$, the structure intervention distance (SID) of the OT algorithm is 329.7 smaller than the one obtained by CAM, which indicates that the graph estimated by the OT algorithm misses fewer edges compared with CAM.

翻译：基于评分函数的贝叶斯网络学习方法旨在最大化全局评分函数。然而，当局部变量同时存在直接依赖和间接依赖时，全局评分优化会遗漏具有间接依赖关系的变量之间的边，因为这些边的评分低于直接依赖关系的边。本文提出了一种基于确定父变量子集的可识别性条件来识别底层有向无环图。基于该可识别性条件，我们开发了一种两阶段算法——最优调优算法，用于对全局优化进行局部修正。在最优阶段，基于一阶希尔伯特-施密特独立性准则的优化问题可得到初始骨架作为初始确定的父变量子集。在调优阶段，利用已从理论上证明的高阶HSIC增量性质，通过删除、添加和DAG形式化策略对骨架进行局部调优。针对不同合成数据集和真实数据集的数值实验表明，OT算法优于现有方法。特别是在图大小为${\rm\bf d=40}$的Sigmoid混合模型中，OT算法的结构干预距离比CAM算法小329.7，表明OT算法估计的图相比CAM遗漏的边更少。