# Stable Structure Learning with HC-Stable and Tabu-Stable Algorithms

arXiv: https://arxiv.org/abs/2504.01740 · HTML: https://mlsanity.com/paper/2504.01740/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 38 references that are in the map.
- First posted on arXiv 2 Apr 2025.

## Builds on (most cited first, 10 of 10)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [A survey of Bayesian Network structure learning](https://mlsanity.com/paper/2109.11415/) (2021)
- [A Comprehensive Scoping Review of Bayesian Networks in Healthcare: Past, Present and Future](https://mlsanity.com/paper/2002.08627/) (2020)
- [A hybrid algorithm for Bayesian network structure learning with application to multi-label learning](https://mlsanity.com/paper/1506.05692/) (2015)
- [Evaluating structure learning algorithms with a balanced scoring function](https://mlsanity.com/paper/1905.12666/) (2019)
- [bnRep: A repository of Bayesian networks from the academic literature](https://mlsanity.com/paper/2409.19158/) (2024)
- [Open problems in causal structure learning: A case study of COVID-19 in the UK](https://mlsanity.com/paper/2305.03859/) (2023)
- [The Impact of Variable Ordering on Bayesian Network Structure Learning](https://mlsanity.com/paper/2206.08952/) (2022)
- [Kutato: An Entropy-Driven System for Construction of Probabilistic Expert Systems from Databases](https://mlsanity.com/paper/1304.1088/) (2013)

## Built on by (most cited first, 2 of 2)

- [LLM-Augmented Causal Discovery: Probabilistic Fusion of Edge Existence and Orientation](https://mlsanity.com/paper/2608.27472/) (2026)
- [Copula Adapted Directed Acyclic Graph for Cluster Representation of Biomedical Data](https://mlsanity.com/paper/2609.16240/) (2026)

## Abstract (arXiv)

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.