This work introduces a novel technique, named structural dimension reduction, to collapse a Bayesian network onto a minimum and localized one while ensuring that probabilistic inferences between the original and reduced networks remain consistent. To this end, we propose a new combinatorial structure in directed acyclic graphs called the directed convex hull, which has turned out to be equivalent to their minimum localized Bayesian networks. An efficient polynomial-time algorithm is devised to identify them by determining the unique directed convex hulls containing the variables of interest from the original networks. Experiments demonstrate that the proposed technique has high dimension reduction capability in real networks, and the efficiency of probabilistic inference based on directed convex hulls can be significantly improved compared with traditional methods such as variable elimination and belief propagation algorithms. The code of this study is open at \href{https://github.com/Balance-H/Algorithms}{https://github.com/Balance-H/Algorithms} and the proofs of the results in the main body are postponed to the appendix.

翻译：本研究提出了一种名为结构降维的新技术，旨在将贝叶斯网络坍缩为一个最小且局部化的网络，同时确保原始网络与降维网络之间的概率推理保持一致。为此，我们在有向无环图中提出了一种新的组合结构，称为有向凸包，该结构被证明等价于其最小局部化贝叶斯网络。我们设计了一种高效的多项式时间算法，通过从原始网络中确定包含感兴趣变量的唯一有向凸包来识别它们。实验表明，所提出的技术在真实网络中具有较高的降维能力，并且基于有向凸包的概率推理效率相较于变量消除、置信传播等传统方法可得到显著提升。本研究的代码公开于 \href{https://github.com/Balance-H/Algorithms}{https://github.com/Balance-H/Algorithms}，正文中结果的证明详见附录。