Learning Probabilities of Causation with Mask-Augmented Data

Probabilities of causation play a central role in modern decision making. Tian and Pearl first introduced formal definitions and derived tight bounds for three binary probabilities of causation, such as the probability of necessity and sufficiency (PNS). However, estimating these probabilities requires both experimental and observational distributions specific to each subpopulation, which are often unreliable or impractical to obtain from limited population-level data. To solve this problem, we propose two machine learning models: Exact-MLP and Mask-MLP, which are trained on a small set of reliable subpopulations and are able to predict PNS bounds for all other subpopulations. We validate our models across four Structural Causal Models (SCMs), each evaluated on population-level data with sample sizes between 100k and 200k. Our models achieve average mean absolute errors (MAEs) of roughly 0.03 on main tasks, reducing MAE by about 80% relative to the corresponding baselines. These results demonstrate both the feasibility of machine learning models for learning probabilities of causation and the effectiveness of the proposed approach.