# Probabilities of Causation: Bounds and Identification

arXiv: https://arxiv.org/abs/1301.3898 · HTML: https://mlsanity.com/paper/1301.3898/

## Key facts

- Cited by 90 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2013.

## Built on by (most cited first, 30 of 90)

- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [Explaining Black-Box Algorithms Using Probabilistic Contrastive Counterfactuals](https://mlsanity.com/paper/2103.11972/) (2021)
- [Local Explanations via Necessity and Sufficiency: Unifying Theory and Practice](https://mlsanity.com/paper/2103.14651/) (2021)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Causes of Effects: Learning individual responses from population data](https://mlsanity.com/paper/2104.13730/) (2021)
- [Personalized Decision Making -- A Conceptual Introduction](https://mlsanity.com/paper/2208.09558/) (2022)
- [Counterfactual harm](https://mlsanity.com/paper/2204.12993/) (2022)
- [Causal Concept Graph Models: Beyond Causal Opacity in Deep Learning](https://mlsanity.com/paper/2405.16507/) (2024)
- [Probabilities of Causation with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09568/) (2022)
- [Causal Inference Through the Structural Causal Marginal Problem](https://mlsanity.com/paper/2202.01300/) (2022)
- [Bounds on Causal Effects and Application to High Dimensional Data](https://mlsanity.com/paper/2106.12121/) (2021)
- [Estimating Categorical Counterfactuals via Deep Twin Networks](https://mlsanity.com/paper/2109.01904/) (2021)
- [Unit Selection with Causal Diagram](https://mlsanity.com/paper/2109.07556/) (2021)
- [On Learning Necessary and Sufficient Causal Graphs](https://mlsanity.com/paper/2301.12389/) (2023)
- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [On the Probability of Necessity and Sufficiency of Explaining Graph Neural Networks: A Lower Bound Optimization Approach](https://mlsanity.com/paper/2212.07056/) (2022)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Counterfactual Analysis in Dynamic Latent State Models](https://mlsanity.com/paper/2205.13832/) (2022)
- [Approximating Counterfactual Bounds while Fusing Observational, Biased and Randomised Data Sources](https://mlsanity.com/paper/2307.16577/) (2023)
- [Probabilities of Causation: Adequate Size of Experimental and Observational Samples](https://mlsanity.com/paper/2210.05027/) (2022)
- [On the Complexity of Counterfactual Reasoning](https://mlsanity.com/paper/2211.13447/) (2022)
- [Reconciling Predictive and Statistical Parity: A Causal Approach](https://mlsanity.com/paper/2306.05059/) (2023)
- [Confounding Robust Deep Reinforcement Learning: A Causal Approach](https://mlsanity.com/paper/2510.21110/) (2025)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [Unit Selection: Case Study and Comparison with A/B Test Heuristic](https://mlsanity.com/paper/2210.05030/) (2022)
- [Probabilities of Causation: Role of Observational Data](https://mlsanity.com/paper/2210.08874/) (2022)
- [Epsilon-Identifiability of Causal Quantities](https://mlsanity.com/paper/2301.12022/) (2023)
- [CausalARC: Abstract Reasoning with Causal World Models](https://mlsanity.com/paper/2509.03636/) (2025)
- [SBI: A Simulation-Based Test of Identifiability for Bayesian Causal Inference](https://mlsanity.com/paper/2102.11761/) (2021)

## Abstract (arXiv)

This paper deals with the problem of estimating the probability that one event was a cause of another in a given scenario. Using structural-semantical definitions of the probabilities of necessary or sufficient causation (or both), we show how to optimally bound these quantities from data obtained in experimental and observational studies, making minimal assumptions concerning the data-generating process. In particular, we strengthen the results of Pearl (1999) by weakening the data-generation assumptions and deriving theoretically sharp bounds on the probabilities of causation. These results delineate precisely how empirical data can be used both in settling questions of attribution and in solving attribution-related problems of decision making.