# Probabilities of Causation with Nonbinary Treatment and Effect

arXiv: https://arxiv.org/abs/2208.09568 · HTML: https://mlsanity.com/paper/2208.09568/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 13 references that are in the map.
- First posted on arXiv 19 Aug 2022.

## Builds on (most cited first, 5 of 5)

- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [Causes of Effects: Learning individual responses from population data](https://mlsanity.com/paper/2104.13730/) (2021)
- [Bounds on Causal Effects and Application to High Dimensional Data](https://mlsanity.com/paper/2106.12121/) (2021)
- [Unit Selection with Causal Diagram](https://mlsanity.com/paper/2109.07556/) (2021)

## Built on by (most cited first, 24 of 24)

- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [Unit Selection with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09569/) (2022)
- [Probabilities of Causation: Adequate Size of Experimental and Observational Samples](https://mlsanity.com/paper/2210.05027/) (2022)
- [Probabilities of Causation for Continuous and Vector Variables](https://mlsanity.com/paper/2405.20487/) (2024)
- [Unit Selection: Learning Benefit Function from Finite Population Data](https://mlsanity.com/paper/2210.08203/) (2022)
- [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)
- [Bounding the Probabilities of Benefit and Harm Through Sensitivity Parameters and Proxies](https://mlsanity.com/paper/2303.05396/) (2023)
- [Bounding probabilities of causation through the causal marginal problem](https://mlsanity.com/paper/2304.02023/) (2023)
- [Moments of Causal Effects](https://mlsanity.com/paper/2505.04971/) (2025)
- [Decomposition of Probabilities of Causation with Two Mediators](https://mlsanity.com/paper/2505.04983/) (2025)
- [Unifying Invariant and Variant Features for Graph Out-of-Distribution via Probability of Necessity and Sufficiency](https://mlsanity.com/paper/2407.15273/) (2024)
- [Estimating Probabilities of Causation with Machine Learning Models](https://mlsanity.com/paper/2502.08858/) (2025)
- [Statistical Decision Theory with Counterfactual Loss](https://mlsanity.com/paper/2505.08908/) (2025)
- [Tightening Bounds on Probabilities of Causation By Merging Datasets](https://mlsanity.com/paper/2310.08406/) (2023)
- [Identification of Probabilities of Causation: from Recursive to Closed-Form Bounds](https://mlsanity.com/paper/2505.15274/) (2025)
- [Learning Probabilities of Causation with Mask-Augmented Data](https://mlsanity.com/paper/2505.17133/) (2025)
- [Preference-based Conditional Treatment Effects and Policy Learning](https://mlsanity.com/paper/2602.03823/) (2026)
- [Bounding Probabilities of Causation with Partial Causal Diagrams](https://mlsanity.com/paper/2602.14503/) (2026)
- [Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions](https://mlsanity.com/paper/2602.18762/) (2026)
- [Causality-Based Feature Importance Quantifying Methods: PN-FI, PS-FI and PNS-FI](https://mlsanity.com/paper/2308.14474/) (2023)
- [General sample size analysis for probabilities of causation: a delta method approach](https://mlsanity.com/paper/2602.17070/) (2026)
- [Causal EpiNets: Precision-corrected Bounds on Individual Treatment Effects using Epistemic Neural Networks](https://mlsanity.com/paper/2605.07065/) (2026)

## Abstract (arXiv)

This paper deals with the problem of estimating the probabilities of causation when treatment and effect are not binary. Tian and Pearl derived sharp bounds for the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of necessity (PN) using experimental and observational data. In this paper, we provide theoretical bounds for all types of probabilities of causation to multivalued treatments and effects. We further discuss examples where our bounds guide practical decisions and use simulation studies to evaluate how informative the bounds are for various combinations of data.