# Probabilities of Causation for Continuous and Vector Variables

arXiv: https://arxiv.org/abs/2405.20487 · HTML: https://mlsanity.com/paper/2405.20487/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 53 references that are in the map.
- First posted on arXiv 30 May 2024.

## Builds on (most cited first, 17 of 17)

- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Learning Deep Features in Instrumental Variable Regression](https://mlsanity.com/paper/2010.07154/) (2020)
- [Nonparametric methods for doubly robust estimation of continuous treatment effects](https://mlsanity.com/paper/1507.00747/) (2015)
- [Policy Evaluation and Optimization with Continuous Treatments](https://mlsanity.com/paper/1802.06037/) (2018)
- [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)
- [Contextual Bandits with Continuous Actions: Smoothing, Zooming, and Adapting](https://mlsanity.com/paper/1902.01520/) (2019)
- [Causes of Effects: Learning individual responses from population data](https://mlsanity.com/paper/2104.13730/) (2021)
- [Efficient Contextual Bandits with Continuous Actions](https://mlsanity.com/paper/2006.06040/) (2020)
- [Probabilities of Causation with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09568/) (2022)
- [Unit Selection with Causal Diagram](https://mlsanity.com/paper/2109.07556/) (2021)
- [Unit Selection with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09569/) (2022)
- [End-to-End Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2107.13068/) (2021)
- [Estimating scaled treatment effects with multiple outcomes](https://mlsanity.com/paper/1608.02273/) (2016)
- [Probabilities of Causation: Role of Observational Data](https://mlsanity.com/paper/2210.08874/) (2022)
- [Bounding the Probability of Causation in Mediation Analysis](https://mlsanity.com/paper/1411.2636/) (2014)
- [From Statistical Evidence to Evidence of Causality](https://mlsanity.com/paper/1311.7513/) (2013)

## Built on by (most cited first, 8 of 8)

- [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)
- [Mediation Analysis for Probabilities of Causation](https://mlsanity.com/paper/2412.14491/) (2024)
- [Preference-based Conditional Treatment Effects and Policy Learning](https://mlsanity.com/paper/2602.03823/) (2026)
- [Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions](https://mlsanity.com/paper/2602.18762/) (2026)
- [Removing Spurious Correlation from Neural Network Interpretations](https://mlsanity.com/paper/2412.02893/) (2024)
- [Causal EpiNets: Precision-corrected Bounds on Individual Treatment Effects using Epistemic Neural Networks](https://mlsanity.com/paper/2605.07065/) (2026)
- [Partial Identification under Causal Orders by Linear Programming](https://mlsanity.com/paper/2608.24427/) (2026)

## Abstract (arXiv)

Probabilities of causation (PoC) are valuable concepts for explainable artificial intelligence and practical decision-making. PoC are originally defined for scalar binary variables. In this paper, we extend the concept of PoC to continuous treatment and outcome variables, and further generalize PoC to capture causal effects between multiple treatments and multiple outcomes. In addition, we consider PoC for a sub-population and PoC with multi-hypothetical terms to capture more sophisticated counterfactual information useful for decision-making. We provide a nonparametric identification theorem for each type of PoC we introduce. Finally, we illustrate the application of our results on a real-world dataset about education.