# A survey of Bayesian Network structure learning

arXiv: https://arxiv.org/abs/2109.11415 · HTML: https://mlsanity.com/paper/2109.11415/

## Key facts

- Cited by 87 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 222 references that are in the map.
- First posted on arXiv 23 Sep 2021.

## Builds on (most cited first, 30 of 51)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Causal Interpretability for Machine Learning -- Problems, Methods and Evaluation](https://mlsanity.com/paper/2003.03934/) (2020)
- [Learning Equivalence Classes of Bayesian Networks Structures](https://mlsanity.com/paper/1302.3566/) (2013)
- [Learning Gaussian Networks](https://mlsanity.com/paper/1302.6808/) (2013)
- [Learning Sparse Causal Models is not NP-hard](https://mlsanity.com/paper/1309.6824/) (2013)
- [Constraint-based Causal Discovery from Multiple Interventions over Overlapping Variable Sets](https://mlsanity.com/paper/1403.2150/) (2014)
- [Causal Discovery in the Presence of Missing Data](https://mlsanity.com/paper/1807.04010/) (2018)
- [High-dimensional consistency in score-based and hybrid structure learning](https://mlsanity.com/paper/1507.02608/) (2015)
- [Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions](https://mlsanity.com/paper/1301.6697/) (2013)
- [The impact of prior knowledge on causal structure learning](https://mlsanity.com/paper/2102.00473/) (2021)
- [Addendum on the scoring of Gaussian directed acyclic graphical models](https://mlsanity.com/paper/1402.6863/) (2014)
- [Partition MCMC for inference on acyclic digraphs](https://mlsanity.com/paper/1504.05006/) (2015)
- [Towards Scalable Bayesian Learning of Causal DAGs](https://mlsanity.com/paper/2010.00684/) (2020)
- [Large-scale empirical validation of Bayesian Network structure learning algorithms with noisy data](https://mlsanity.com/paper/2005.09020/) (2020)
- [Theory Refinement on Bayesian Networks](https://mlsanity.com/paper/1303.5709/) (2013)
- [Efficient Sampling and Structure Learning of Bayesian Networks](https://mlsanity.com/paper/1803.07859/) (2018)
- [Ordering-Based Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/1910.09014/) (2019)
- [Fast Causal Inference with Non-Random Missingness by Test-Wise Deletion](https://mlsanity.com/paper/1705.09031/) (2017)
- [Learning Bayesian Network Structure from Massive Datasets: The "Sparse Candidate" Algorithm](https://mlsanity.com/paper/1301.6696/) (2013)
- [Bayesian structure learning and sampling of Bayesian networks with the R package BiDAG](https://mlsanity.com/paper/2105.00488/) (2021)
- [Structural Intervention Distance (SID) for Evaluating Causal Graphs](https://mlsanity.com/paper/1306.1043/) (2013)
- [Properties of Bayesian Belief Network Learning Algorithms](https://mlsanity.com/paper/1302.6792/) (2013)
- [Greedy structure learning from data that contain systematic missing values](https://mlsanity.com/paper/2107.04184/) (2021)
- [A hybrid algorithm for Bayesian network structure learning with application to multi-label learning](https://mlsanity.com/paper/1506.05692/) (2015)
- [Improving Accuracy and Scalability of the PC Algorithm by Maximizing P-value](https://mlsanity.com/paper/1610.00378/) (2016)
- [A Uniformly Consistent Estimator of Causal Effects under the k-Triangle-Faithfulness Assumption](https://mlsanity.com/paper/1502.00829/) (2015)

## Built on by (most cited first, 30 of 87)

- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [The Essential Role of Causality in Foundation World Models for Embodied AI](https://mlsanity.com/paper/2402.06665/) (2024)
- [A Scale-Invariant Sorting Criterion to Find a Causal Order in Additive Noise Models](https://mlsanity.com/paper/2303.18211/) (2023)
- [Causal Structure Learning Supervised by Large Language Model](https://mlsanity.com/paper/2311.11689/) (2023)
- [FedDAG: Federated DAG Structure Learning](https://mlsanity.com/paper/2112.03555/) (2021)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Local Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/2405.16225/) (2024)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)
- [bnRep: A repository of Bayesian networks from the academic literature](https://mlsanity.com/paper/2409.19158/) (2024)
- [Effective and efficient structure learning with pruning and model averaging strategies](https://mlsanity.com/paper/2112.00398/) (2021)
- [Open problems in causal structure learning: A case study of COVID-19 in the UK](https://mlsanity.com/paper/2305.03859/) (2023)
- [Adaptive Stream Processing on Edge Devices through Active Inference](https://mlsanity.com/paper/2409.17937/) (2024)
- [Machine learning interatomic potential can infer electrical response](https://mlsanity.com/paper/2504.05169/) (2025)
- [The Impact of Variable Ordering on Bayesian Network Structure Learning](https://mlsanity.com/paper/2206.08952/) (2022)
- [Mitigating Prior Errors in Causal Structure Learning: A Resilient Approach via Bayesian Networks](https://mlsanity.com/paper/2306.07032/) (2023)
- [Causal Inference for Banking Finance and Insurance A Survey](https://mlsanity.com/paper/2307.16427/) (2023)
- [Enhancing Interpretability and Generalizability in Extended Isolation Forests](https://mlsanity.com/paper/2310.05468/) (2023)
- [Learning bounded-degree polytrees with known skeleton](https://mlsanity.com/paper/2310.06333/) (2023)
- [Causal discovery using dynamically requested knowledge](https://mlsanity.com/paper/2310.11154/) (2023)
- [A Bayesian Take on Gaussian Process Networks](https://mlsanity.com/paper/2306.11380/) (2023)
- [ProDAG: Projected Variational Inference for Directed Acyclic Graphs](https://mlsanity.com/paper/2405.15167/) (2024)
- [Investigating potential causes of Sepsis with Bayesian network structure learning](https://mlsanity.com/paper/2406.09207/) (2024)
- [Learning Discrete Directed Acyclic Graphs via Backpropagation](https://mlsanity.com/paper/2210.15353/) (2022)
- [Poisoning Network Flow Classifiers](https://mlsanity.com/paper/2306.01655/) (2023)
- [Using GPT-4 to guide causal machine learning](https://mlsanity.com/paper/2407.18607/) (2024)
- [Adaptive Active Inference Agents for Heterogeneous and Lifelong Federated Learning](https://mlsanity.com/paper/2410.09099/) (2024)
- [Discovery and density estimation of latent confounders in Bayesian networks with evidence lower bound](https://mlsanity.com/paper/2206.05490/) (2022)
- [Weakly-supervised causal discovery based on fuzzy knowledge and complex data complementarity](https://mlsanity.com/paper/2405.08699/) (2024)
- [Foundations of a Developmental Design Paradigm for Integrated Continual Learning, Deliberative Behavior, and Comprehensibility](https://mlsanity.com/paper/2502.13935/) (2025)
- [Stable Structure Learning with HC-Stable and Tabu-Stable Algorithms](https://mlsanity.com/paper/2504.01740/) (2025)

## Abstract (arXiv)

Bayesian Networks (BNs) have become increasingly popular over the last few decades as a tool for reasoning under uncertainty in fields as diverse as medicine, biology, epidemiology, economics and the social sciences. This is especially true in real-world areas where we seek to answer complex questions based on hypothetical evidence to determine actions for intervention. However, determining the graphical structure of a BN remains a major challenge, especially when modelling a problem under causal assumptions. Solutions to this problem include the automated discovery of BN graphs from data, constructing them based on expert knowledge, or a combination of the two. This paper provides a comprehensive review of combinatoric algorithms proposed for learning BN structure from data, describing 74 algorithms including prototypical, well-established and state-of-the-art approaches. The basic approach of each algorithm is described in consistent terms, and the similarities and differences between them highlighted. Methods of evaluating algorithms and their comparative performance are discussed including the consistency of claims made in the literature. Approaches for dealing with data noise in real-world datasets and incorporating expert knowledge into the learning process are also covered.