# On Learning Necessary and Sufficient Causal Graphs

arXiv: https://arxiv.org/abs/2301.12389 · HTML: https://mlsanity.com/paper/2301.12389/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 29 Jan 2023.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Counterfactual VQA: A Cause-Effect Look at Language Bias](https://mlsanity.com/paper/2006.04315/) (2020)
- [Long-Tailed Classification by Keeping the Good and Removing the Bad Momentum Causal Effect](https://mlsanity.com/paper/2009.12991/) (2020)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Causal Inference in Natural Language Processing: Estimation, Prediction, Interpretation and Beyond](https://mlsanity.com/paper/2109.00725/) (2021)
- [Feature relevance quantification in explainable AI: A causal problem](https://mlsanity.com/paper/1910.13413/) (2019)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [Estimating the effect of joint interventions from observational data in sparse high-dimensional settings](https://mlsanity.com/paper/1407.2451/) (2014)
- [Inference for Individual Mediation Effects and Interventional Effects in Sparse High-Dimensional Causal Graphical Models](https://mlsanity.com/paper/1809.10652/) (2018)
- [Testing Mediation Effects Using Logic of Boolean Matrices](https://mlsanity.com/paper/2005.04584/) (2020)

## Built on by (most cited first, 14 of 14)

- [Towards Trustworthy Explanation: On Causal Rationalization](https://mlsanity.com/paper/2306.14115/) (2023)
- [Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure-Outcome Pairs](https://mlsanity.com/paper/2310.17816/) (2023)
- [Local Causal Discovery for Structural Evidence of Direct Discrimination](https://mlsanity.com/paper/2405.14848/) (2024)
- [Compositional Causal Reasoning Evaluation in Language Models](https://mlsanity.com/paper/2503.04556/) (2025)
- [Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairness](https://mlsanity.com/paper/2408.03425/) (2024)
- [Image Quality Assessment: Investigating Causal Perceptual Effects with Abductive Counterfactual Inference](https://mlsanity.com/paper/2412.16939/) (2024)
- [Unifying Invariance and Spuriousity for Graph Out-of-Distribution via Probability of Necessity and Sufficiency](https://mlsanity.com/paper/2402.09165/) (2024)
- [Unifying Invariant and Variant Features for Graph Out-of-Distribution via Probability of Necessity and Sufficiency](https://mlsanity.com/paper/2407.15273/) (2024)
- [Text Rationalization for Robust Causal Effect Estimation](https://mlsanity.com/paper/2512.05373/) (2025)
- [Causal Perception Inspired Representation Learning for Trustworthy Image Quality Assessment](https://mlsanity.com/paper/2404.19567/) (2024)
- [Seeking the Sufficiency and Necessity Causal Features in Multimodal Representation Learning](https://mlsanity.com/paper/2408.16577/) (2024)
- [Where to Intervene: Action Selection in Deep Reinforcement Learning](https://mlsanity.com/paper/2507.04187/) (2025)
- [ReBeCA: Unveiling Interpretable Behavior Hierarchy behind the Iterative Self-Reflection of Language Models with Causal Analysis](https://mlsanity.com/paper/2602.06373/) (2026)
- [Seeking Necessary and Sufficient Information from Multimodal Medical Data](https://mlsanity.com/paper/2603.00289/) (2026)

## Abstract (arXiv)

The causal revolution has stimulated interest in understanding complex relationships in various fields. Most of the existing methods aim to discover causal relationships among all variables within a complex large-scale graph. However, in practice, only a small subset of variables in the graph are relevant to the outcomes of interest. Consequently, causal estimation with the full causal graph -- particularly given limited data -- could lead to numerous falsely discovered, spurious variables that exhibit high correlation with, but exert no causal impact on, the target outcome. In this paper, we propose learning a class of necessary and sufficient causal graphs (NSCG) that exclusively comprises causally relevant variables for an outcome of interest, which we term causal features. The key idea is to employ probabilities of causation to systematically evaluate the importance of features in the causal graph, allowing us to identify a subgraph relevant to the outcome of interest. To learn NSCG from data, we develop a necessary and sufficient causal structural learning (NSCSL) algorithm, by establishing theoretical properties and relationships between probabilities of causation and natural causal effects of features. Across empirical studies of simulated and real data, we demonstrate that NSCSL outperforms existing algorithms and can reveal crucial yeast genes for target heritable traits of interest.