Many Bayesian Network structure learning algorithms are unstable, with the learned graph sensitive to arbitrary dataset artifacts, such as the ordering of columns (i.e., variable order). PC-Stable attempts to address this issue for the widely-used PC algorithm, prompting researchers to use the "stable" version instead. However, this problem seems to have been overlooked for score-based algorithms. In this study, we show that some widely-used score-based algorithms, as well as hybrid and constraint-based algorithms, including PC-Stable, suffer from the same issue. We propose a novel solution for score-based greedy hill-climbing that eliminates instability by determining a stable node order, leading to consistent results regardless of variable ordering. Two implementations, HC-Stable and Tabu-Stable, are introduced. Tabu-Stable achieves the highest BIC scores across all networks, and the highest accuracy for categorical networks. These results highlight the importance of addressing instability in structure learning and provide a robust and practical approach for future applications. This extends the scope and impact of our previous work presented at Probabilistic Graphical Models 2024 by incorporating continuous variables. The implementation, along with usage instructions, is freely available on GitHub at https://github.com/causal-iq/discovery.

翻译：许多贝叶斯网络结构学习算法具有不稳定性，其学习得到的图对数据集中的人为因素（例如列的顺序，即变量顺序）十分敏感。PC-Stable尝试为广泛使用的PC算法解决此问题，促使研究者转而使用其“稳定”版本。然而，这一问题在基于得分的算法中似乎被忽视了。在本研究中，我们证明一些广泛使用的基于得分的算法，以及混合算法和基于约束的算法（包括PC-Stable），均存在相同的问题。我们为基于得分的贪心爬山法提出了一种新颖的解决方案，通过确定稳定的节点顺序来消除不稳定性，从而确保无论变量顺序如何都能获得一致的结果。我们引入了两种实现：HC-Stable与Tabu-Stable。Tabu-Stable在所有网络上均获得了最高的BIC得分，并在分类网络上取得了最高的准确率。这些结果突显了解决结构学习中不稳定性问题的重要性，并为未来的应用提供了一种稳健且实用的方法。本研究通过纳入连续变量，扩展了我们先前在Probabilistic Graphical Models 2024会议上发表工作的范围和影响力。相关实现及使用说明已在GitHub上开源，地址为：https://github.com/causal-iq/discovery。