# Integer Programming for Causal Structure Learning in the Presence of Latent Variables

arXiv: https://arxiv.org/abs/2102.03129 · HTML: https://mlsanity.com/paper/2102.03129/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 35 references that are in the map.
- First posted on arXiv 5 Feb 2021.

## Builds on (most cited first, 9 of 9)

- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Differentiable Causal Discovery Under Unmeasured Confounding](https://mlsanity.com/paper/2010.06978/) (2020)
- [Ordering-Based Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/1910.09014/) (2019)
- [Integer Programming for Learning Directed Acyclic Graphs from Continuous Data](https://mlsanity.com/paper/1904.10574/) (2019)
- [Bayesian Network Learning via Topological Order](https://mlsanity.com/paper/1701.05654/) (2017)
- [Markovian acyclic directed mixed graphs for discrete data](https://mlsanity.com/paper/1301.6624/) (2013)
- [A factorization criterion for acyclic directed mixed graphs](https://mlsanity.com/paper/1406.6764/) (2014)
- [Distributional Equivalence and Structure Learning for Bow-free Acyclic Path Diagrams](https://mlsanity.com/paper/1508.01717/) (2015)
- [Bayesian network structure learning with causal effects in the presence of latent variables](https://mlsanity.com/paper/2005.14381/) (2020)

## Built on by (most cited first, 15 of 15)

- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Local Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/2405.16225/) (2024)
- [Scalable Differentiable Causal Discovery in the Presence of Latent Confounders with Skeleton Posterior (Extended Version)](https://mlsanity.com/paper/2406.10537/) (2024)
- [Learning Cut Generating Functions for Integer Programming](https://mlsanity.com/paper/2405.13992/) (2024)
- [The m-connecting imset and factorization for ADMG models](https://mlsanity.com/paper/2207.08963/) (2022)
- [Taming Binarized Neural Networks and Mixed-Integer Programs](https://mlsanity.com/paper/2310.04469/) (2023)
- [Differentiable Cyclic Causal Discovery Under Unmeasured Confounders](https://mlsanity.com/paper/2508.08450/) (2025)
- [A fast score-based search algorithm for maximal ancestral graphs using entropy](https://mlsanity.com/paper/2402.04777/) (2024)
- [ExDAG: an MIQP Algorithm for Learning DAGs](https://mlsanity.com/paper/2406.15229/) (2024)
- [ExMAG: Learning of Maximally Ancestral Graphs](https://mlsanity.com/paper/2503.08245/) (2025)
- [Testing for Causal Fairness](https://mlsanity.com/paper/2502.12874/) (2025)
- [Inexact Column Generation for Bayesian Network Structure Learning via Difference-of-Submodular Optimization](https://mlsanity.com/paper/2505.11089/) (2025)
- [Efficient Differentiable Causal Discovery via Reliable Super-Structure Learning](https://mlsanity.com/paper/2601.05474/) (2026)
- [From Local to Cluster: A Unified Framework for Causal Discovery with Latent Variables](https://mlsanity.com/paper/2604.22416/) (2026)
- [DAG-CLIP: A DAG Learning Framework in the Presence of Latent Variables](https://mlsanity.com/paper/2610.06086/) (2026)

## Abstract (arXiv)

The problem of finding an ancestral acyclic directed mixed graph (ADMG) that represents the causal relationships between a set of variables is an important area of research on causal inference. Most existing score-based structure learning methods focus on learning directed acyclic graph (DAG) models without latent variables. A number of score-based methods have recently been proposed for the ADMG learning, yet they are heuristic in nature and do not guarantee an optimal solution. We propose a novel exact score-based method that solves an integer programming (IP) formulation and returns a score-maximizing ancestral ADMG for a set of continuous variables that follow a multivariate Gaussian distribution. We generalize the state-of-the-art IP model for DAG learning problems and derive new classes of valid inequalities to formulate an IP model for ADMG learning. Empirically, our model can be solved efficiently for medium-sized problems and achieves better accuracy than state-of-the-art score-based methods as well as benchmark constraint-based methods.