# Strong Completeness and Faithfulness in Bayesian Networks

arXiv: https://arxiv.org/abs/1302.4973 · HTML: https://mlsanity.com/paper/1302.4973/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2013.

## Built on by (most cited first, 30 of 45)

- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Robust agents learn causal world models](https://mlsanity.com/paper/2402.10877/) (2024)
- [Explaining Predictive Uncertainty with Information Theoretic Shapley Values](https://mlsanity.com/paper/2306.05724/) (2023)
- [Scientific Inference With Interpretable Machine Learning: Analyzing Models to Learn About Real-World Phenomena](https://mlsanity.com/paper/2206.05487/) (2022)
- [Causal Deep Learning](https://mlsanity.com/paper/2303.02186/) (2023)
- [Invariant Policy Learning: A Causal Perspective](https://mlsanity.com/paper/2106.00808/) (2021)
- [Selection, Ignorability and Challenges With Causal Fairness](https://mlsanity.com/paper/2202.13774/) (2022)
- [Distributed Learning of Generalized Linear Causal Networks](https://mlsanity.com/paper/2201.09194/) (2022)
- [Detecting hidden confounding in observational data using multiple environments](https://mlsanity.com/paper/2205.13935/) (2022)
- [Subset verification and search algorithms for causal DAGs](https://mlsanity.com/paper/2301.03180/) (2023)
- [ML4C: Seeing Causality Through Latent Vicinity](https://mlsanity.com/paper/2110.00637/) (2021)
- [Toward Falsifying Causal Graphs Using a Permutation-Based Test](https://mlsanity.com/paper/2305.09565/) (2023)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Active causal structure learning with advice](https://mlsanity.com/paper/2305.19588/) (2023)
- [Obtaining Causal Information by Merging Datasets with MAXENT](https://mlsanity.com/paper/2107.07640/) (2021)
- [Expert-Aided Causal Discovery of Ancestral Graphs](https://mlsanity.com/paper/2309.12032/) (2023)
- [Towards Bounding Causal Effects under Markov Equivalence](https://mlsanity.com/paper/2311.07259/) (2023)
- [Causality Pursuit from Heterogeneous Environments via Neural Adversarial Invariance Learning](https://mlsanity.com/paper/2405.04715/) (2024)
- [Modularity in Reinforcement Learning via Algorithmic Independence in Credit Assignment](https://mlsanity.com/paper/2106.14993/) (2021)
- [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)
- [Causality and independence in perfectly adapted dynamical systems](https://mlsanity.com/paper/2101.11885/) (2021)
- [Reinterpreting causal discovery as the task of predicting unobserved joint statistics](https://mlsanity.com/paper/2305.06894/) (2023)
- [Neuro-Causal Factor Analysis](https://mlsanity.com/paper/2305.19802/) (2023)
- [Results on Counterfactual Invariance](https://mlsanity.com/paper/2307.08519/) (2023)
- [Learning Identifiable Structures Helps Avoid Bias in DNN-based Supervised Causal Learning](https://mlsanity.com/paper/2502.10883/) (2025)
- [Risk Assessment of Lymph Node Metastases in Endometrial Cancer Patients: A Causal Approach](https://mlsanity.com/paper/2305.10041/) (2023)
- [On the Independencies Hidden in the Structure of a Probabilistic Logic Program](https://mlsanity.com/paper/2308.15865/) (2023)
- [Enabling Runtime Verification of Causal Discovery Algorithms with Automated Conditional Independence Reasoning (Extended Version)](https://mlsanity.com/paper/2309.05264/) (2023)
- [Characterization and Learning of Causal Graphs from Hard Interventions](https://mlsanity.com/paper/2505.01037/) (2025)

## Abstract (arXiv)

A completeness result for d-separation applied to discrete Bayesian networks is presented and it is shown that in a strong measure-theoretic sense almost all discrete distributions for a given network structure are faithful; i.e. the independence facts true of the distribution are all and only those entailed by the network structure.