# Conditions and Assumptions for Constraint-based Causal Structure Learning

arXiv: https://arxiv.org/abs/2103.13521 · HTML: https://mlsanity.com/paper/2103.13521/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 60 references that are in the map.
- First posted on arXiv 24 Mar 2021.

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

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [On the Equivalence of Causal Models](https://mlsanity.com/paper/1304.1108/) (2013)
- [On the Intersection Property of Conditional Independence and its Application to Causal Discovery](https://mlsanity.com/paper/1403.0408/) (2014)
- [Marginalization and Conditioning for LWF Chain Graphs](https://mlsanity.com/paper/1405.7129/) (2014)

## Built on by (most cited first, 3 of 3)

- [Axiomatization of Interventional Probability Distributions](https://mlsanity.com/paper/2305.04479/) (2023)
- [A General Framework on Conditions for Constraint-based Causal Learning](https://mlsanity.com/paper/2408.07575/) (2024)
- [Uncovering Bias Mechanisms in Observational Studies](https://mlsanity.com/paper/2506.01191/) (2025)

## Abstract (arXiv)

We formalize constraint-based structure learning of the "true" causal graph from observed data when unobserved variables are also existent. We provide conditions for a "natural" family of constraint-based structure-learning algorithms that output graphs that are Markov equivalent to the causal graph. Under the faithfulness assumption, this natural family contains all exact structure-learning algorithms. We also provide a set of assumptions, under which any natural structure-learning algorithm outputs Markov equivalent graphs to the causal graph. These assumptions can be thought of as a relaxation of faithfulness, and most of them can be directly tested from (the underlying distribution) of the data, particularly when one focuses on structural causal models. We specialize the definitions and results for structural causal models.