# Probabilities of Causation: Adequate Size of Experimental and Observational Samples

arXiv: https://arxiv.org/abs/2210.05027 · HTML: https://mlsanity.com/paper/2210.05027/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 10 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [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)
- [Probabilities of Causation with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09568/) (2022)
- [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)
- [Unit Selection with Nonbinary Treatment and Effect](https://mlsanity.com/paper/2208.09569/) (2022)

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

- [Unit Selection: Learning Benefit Function from Finite Population Data](https://mlsanity.com/paper/2210.08203/) (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)
- [Estimating Probabilities of Causation with Machine Learning Models](https://mlsanity.com/paper/2502.08858/) (2025)
- [Learning Probabilities of Causation with Mask-Augmented Data](https://mlsanity.com/paper/2505.17133/) (2025)
- [Causality in the Can: Diet Coke's Impact on Fatness](https://mlsanity.com/paper/2405.10746/) (2024)
- [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)

The probabilities of causation are commonly used to solve decision-making problems. 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. The assumption is that one is in possession of a large enough sample to permit an accurate estimation of the experimental and observational distributions. In this study, we present a method for determining the sample size needed for such estimation, when a given confidence interval (CI) is specified. We further show by simulation that the proposed sample size delivered stable estimations of the bounds of PNS.