# Recursive Causal Structure Learning in the Presence of Latent Variables and Selection Bias

arXiv: https://arxiv.org/abs/2110.12036 · HTML: https://mlsanity.com/paper/2110.12036/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [Doubly Robust Off-policy Value Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1511.03722/) (2015)
- [Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1604.00923/) (2016)
- [Learning Sparse Causal Models is not NP-hard](https://mlsanity.com/paper/1309.6824/) (2013)
- [A Recursive Markov Boundary-Based Approach to Causal Structure Learning](https://mlsanity.com/paper/2010.04992/) (2020)
- [A Constraint-Based Algorithm For Causal Discovery with Cycles, Latent Variables and Selection Bias](https://mlsanity.com/paper/1805.02087/) (2018)
- [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, 29 of 29)

- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables](https://mlsanity.com/paper/2312.11001/) (2023)
- [Identification of Nonlinear Latent Hierarchical Models](https://mlsanity.com/paper/2306.07916/) (2023)
- [Recursive Causal Discovery](https://mlsanity.com/paper/2403.09300/) (2024)
- [Local Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/2405.16225/) (2024)
- [Novel Ordering-based Approaches for Causal Structure Learning in the Presence of Unobserved Variables](https://mlsanity.com/paper/2208.06935/) (2022)
- [A Unified Experiment Design Approach for Cyclic and Acyclic Causal Models](https://mlsanity.com/paper/2205.10083/) (2022)
- [On the Parameter Identifiability of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2407.16975/) (2024)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Causal Imitability Under Context-Specific Independence Relations](https://mlsanity.com/paper/2306.00585/) (2023)
- [Efficient Ensemble Conditional Independence Test Framework for Causal Discovery](https://mlsanity.com/paper/2509.21021/) (2025)
- [Causal Effect Identification in Uncertain Causal Networks](https://mlsanity.com/paper/2208.04627/) (2022)
- [s-ID: Causal Effect Identification in a Sub-Population](https://mlsanity.com/paper/2309.02281/) (2023)
- [Differentiable Causal Discovery For Latent Hierarchical Causal Models](https://mlsanity.com/paper/2411.19556/) (2024)
- [Score-Based Causal Discovery of Latent Variable Causal Models](https://mlsanity.com/paper/2605.20396/) (2026)
- [Experimental Design for Causal Effect Identification](https://mlsanity.com/paper/2205.02232/) (2022)
- [GT-CausIn: a novel causal-based insight for traffic prediction](https://mlsanity.com/paper/2212.05782/) (2022)
- [Score-based Greedy Search for Structure Identification of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2510.04378/) (2025)
- [A Review and Roadmap of Deep Causal Model from Different Causal Structures and Representations](https://mlsanity.com/paper/2311.00923/) (2023)
- [On the Three Demons in Causality in Finance: Time Resolution, Nonstationarity, and Latent Factors](https://mlsanity.com/paper/2401.05414/) (2023)
- [Traceable Latent Variable Discovery Based on Multi-Agent Collaboration](https://mlsanity.com/paper/2602.14456/) (2026)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)
- [Learning Linear Gaussian Polytree Models with Interventions](https://mlsanity.com/paper/2311.04636/) (2023)
- [A Critical Review of Predominant Bias in Neural Networks](https://mlsanity.com/paper/2502.11031/) (2025)
- [Uncovering Bias Mechanisms in Observational Studies](https://mlsanity.com/paper/2506.01191/) (2025)
- [A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/2605.10651/) (2026)
- [Causal Modeling of Selection in Evolution](https://mlsanity.com/paper/2606.05689/) (2026)
- [Decoupled Causal Discovery](https://mlsanity.com/paper/2609.23535/) (2026)
- [Hessian Rank Constraint for Learning Structure of Nonlinear Latent Variable Models](https://mlsanity.com/paper/2609.24241/) (2026)

## Abstract (arXiv)

We consider the problem of learning the causal MAG of a system from observational data in the presence of latent variables and selection bias. Constraint-based methods are one of the main approaches for solving this problem, but the existing methods are either computationally impractical when dealing with large graphs or lacking completeness guarantees. We propose a novel computationally efficient recursive constraint-based method that is sound and complete. The key idea of our approach is that at each iteration a specific type of variable is identified and removed. This allows us to learn the structure efficiently and recursively, as this technique reduces both the number of required conditional independence (CI) tests and the size of the conditioning sets. The former substantially reduces the computational complexity, while the latter results in more reliable CI tests. We provide an upper bound on the number of required CI tests in the worst case. To the best of our knowledge, this is the tightest bound in the literature. We further provide a lower bound on the number of CI tests required by any constraint-based method. The upper bound of our proposed approach and the lower bound at most differ by a factor equal to the number of variables in the worst case. We provide experimental results to compare the proposed approach with the state of the art on both synthetic and real-world structures.