# Causal Networks: Semantics and Expressiveness

arXiv: https://arxiv.org/abs/1304.2379 · HTML: https://mlsanity.com/paper/1304.2379/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2013.

## Built on by (most cited first, 30 of 67)

- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [Greedy Relaxations of the Sparsest Permutation Algorithm](https://mlsanity.com/paper/2206.05421/) (2022)
- [Agent Incentives: A Causal Perspective](https://mlsanity.com/paper/2102.01685/) (2021)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Assumption violations in causal discovery and the robustness of score matching](https://mlsanity.com/paper/2310.13387/) (2023)
- [Stochastic Causal Programming for Bounding Treatment Effects](https://mlsanity.com/paper/2202.10806/) (2022)
- [A Causal Bandit Approach to Learning Good Atomic Interventions in Presence of Unobserved Confounders](https://mlsanity.com/paper/2107.02772/) (2021)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [No Fair Lunch: A Causal Perspective on Dataset Bias in Machine Learning for Medical Imaging](https://mlsanity.com/paper/2307.16526/) (2023)
- [Context-Specific Causal Discovery for Categorical Data Using Staged Trees](https://mlsanity.com/paper/2106.04416/) (2021)
- [Finding Valid Adjustments under Non-ignorability with Minimal DAG Knowledge](https://mlsanity.com/paper/2106.11560/) (2021)
- [Why Fair Labels Can Yield Unfair Predictions: Graphical Conditions for Introduced Unfairness](https://mlsanity.com/paper/2202.10816/) (2022)
- [Valid Inference After Causal Discovery](https://mlsanity.com/paper/2208.05949/) (2022)
- [Rethinking Fair Representation Learning for Performance-Sensitive Tasks](https://mlsanity.com/paper/2410.04120/) (2024)
- [Heteroscedastic Causal Structure Learning](https://mlsanity.com/paper/2307.07973/) (2023)
- [Better Simulations for Validating Causal Discovery with the DAG-Adaptation of the Onion Method](https://mlsanity.com/paper/2405.13100/) (2024)
- [Are Bayesian networks typically faithful?](https://mlsanity.com/paper/2410.16004/) (2024)
- [A Complete Criterion for Value of Information in Soluble Influence Diagrams](https://mlsanity.com/paper/2202.11629/) (2022)
- [Order-based Structure Learning with Normalizing Flows](https://mlsanity.com/paper/2308.07480/) (2023)
- [Identifiability Analysis of Linear ODE Systems with Hidden Confounders](https://mlsanity.com/paper/2410.21917/) (2024)
- [Causal Representation Learning from General Environments under Nonparametric Mixing](https://mlsanity.com/paper/2604.23800/) (2026)
- [Selecting Robust Features for Machine Learning Applications using Multidata Causal Discovery](https://mlsanity.com/paper/2304.05294/) (2023)
- [Front-door Adjustment Beyond Markov Equivalence with Limited Graph Knowledge](https://mlsanity.com/paper/2306.11008/) (2023)
- [Confounded Budgeted Causal Bandits](https://mlsanity.com/paper/2401.07578/) (2024)
- [Embracing Discrete Search: A Reasonable Approach to Causal Structure Learning](https://mlsanity.com/paper/2510.04970/) (2025)
- [Conditions and Assumptions for Constraint-based Causal Structure Learning](https://mlsanity.com/paper/2103.13521/) (2021)
- [Algorithmic Causal Effect Identification with causaleffect](https://mlsanity.com/paper/2107.04632/) (2021)

## Abstract (arXiv)

Dependency knowledge of the form "x is independent of y once z is known" invariably obeys the four graphoid axioms, examples include probabilistic and database dependencies. Often, such knowledge can be represented efficiently with graphical structures such as undirected graphs and directed acyclic graphs (DAGs). In this paper we show that the graphical criterion called d-separation is a sound rule for reading independencies from any DAG based on a causal input list drawn from a graphoid. The rule may be extended to cover DAGs that represent functional dependencies as well as conditional dependencies.