# Efficient Bayesian network structure learning via local Markov boundary search

arXiv: https://arxiv.org/abs/2110.06082 · HTML: https://mlsanity.com/paper/2110.06082/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [Learning linear structural equation models in polynomial time and sample complexity](https://mlsanity.com/paper/1707.04673/) (2017)
- [A polynomial-time algorithm for learning nonparametric causal graphs](https://mlsanity.com/paper/2006.11970/) (2020)
- [Entropic Causal Inference](https://mlsanity.com/paper/1611.04035/) (2016)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [High-Dimensional Causal Discovery Under non-Gaussianity](https://mlsanity.com/paper/1803.11273/) (2018)
- [Learning Bayesian Network Structure from Massive Datasets: The "Sparse Candidate" Algorithm](https://mlsanity.com/paper/1301.6696/) (2013)
- [The Recovery of Causal Poly-Trees from Statistical Data](https://mlsanity.com/paper/1304.2736/) (2013)
- [Learning Quadratic Variance Function (QVF) DAG models via OverDispersion Scoring (ODS)](https://mlsanity.com/paper/1704.08783/) (2017)
- [Optimal rates of entropy estimation over Lipschitz balls](https://mlsanity.com/paper/1711.02141/) (2017)
- [Learning Polytrees](https://mlsanity.com/paper/1301.6688/) (2013)
- [Parameterized Complexity Results for Exact Bayesian Network Structure Learning](https://mlsanity.com/paper/1402.0558/) (2014)
- [On the Sample Complexity of Learning Bayesian Networks](https://mlsanity.com/paper/1302.3579/) (2013)
- [SparsityBoost: A New Scoring Function for Learning Bayesian Network Structure](https://mlsanity.com/paper/1309.6820/) (2013)
- [On the Intersection Property of Conditional Independence and its Application to Causal Discovery](https://mlsanity.com/paper/1403.0408/) (2014)
- [Entropic Causality and Greedy Minimum Entropy Coupling](https://mlsanity.com/paper/1701.08254/) (2017)
- [Learning Bayesian Networks with Low Rank Conditional Probability Tables](https://mlsanity.com/paper/1905.12552/) (2019)

## Built on by (most cited first, 14 of 14)

- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Optimal estimation of Gaussian DAG models](https://mlsanity.com/paper/2201.10548/) (2022)
- [Uncovering Meanings of Embeddings via Partial Orthogonality](https://mlsanity.com/paper/2310.17611/) (2023)
- [A Fast Non-parametric Approach for Local Causal Structure Learning](https://mlsanity.com/paper/2111.14969/) (2021)
- [Structural restrictions in local causal discovery: identifying direct causes of a target variable](https://mlsanity.com/paper/2307.16048/) (2023)
- [Order-based Structure Learning with Normalizing Flows](https://mlsanity.com/paper/2308.07480/) (2023)
- [Learning bounded-degree polytrees with known skeleton](https://mlsanity.com/paper/2310.06333/) (2023)
- [Optimal estimation of Gaussian (poly)trees](https://mlsanity.com/paper/2402.06380/) (2024)
- [Learning Gaussian DAG Models without Condition Number Bounds](https://mlsanity.com/paper/2511.06164/) (2025)
- [Distinguishing Cause from Effect on Categorical Data: The Uniform Channel Model](https://mlsanity.com/paper/2303.08572/) (2023)
- [Causal Layering via Conditional Entropy](https://mlsanity.com/paper/2401.10495/) (2024)
- [Distribution Learning Meets Graph Structure Sampling](https://mlsanity.com/paper/2405.07914/) (2024)
- [Estimating large causal polytrees from small samples](https://mlsanity.com/paper/2209.07028/) (2022)
- [Optimal structure learning and conditional independence testing](https://mlsanity.com/paper/2507.05689/) (2025)

## Abstract (arXiv)

We analyze the complexity of learning directed acyclic graphical models from observational data in general settings without specific distributional assumptions. Our approach is information-theoretic and uses a local Markov boundary search procedure in order to recursively construct ancestral sets in the underlying graphical model. Perhaps surprisingly, we show that for certain graph ensembles, a simple forward greedy search algorithm (i.e. without a backward pruning phase) suffices to learn the Markov boundary of each node. This substantially improves the sample complexity, which we show is at most polynomial in the number of nodes. This is then applied to learn the entire graph under a novel identifiability condition that generalizes existing conditions from the literature. As a matter of independent interest, we establish finite-sample guarantees for the problem of recovering Markov boundaries from data. Moreover, we apply our results to the special case of polytrees, for which the assumptions simplify, and provide explicit conditions under which polytrees are identifiable and learnable in polynomial time. We further illustrate the performance of the algorithm, which is easy to implement, in a simulation study. Our approach is general, works for discrete or continuous distributions without distributional assumptions, and as such sheds light on the minimal assumptions required to efficiently learn the structure of directed graphical models from data.