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Estimating Probabilities of Causation with Machine Learning Models

Shuai Wang, Ang Li

TL;DR

Estimating probabilities of causation for subpopulations with limited data is challenging under Structural Causal Models. The authors propose a machine learning framework to predict the bounds of $PNS$, $PS$, and $PN$ by training on subpopulations with sufficient data, focusing on accurately predicting $PNS$. Evaluating five models, they find that a Multilayer Perceptron with Mish activation delivers the best performance, achieving a mean absolute error near $0.02$ when predicting $PNS$ bounds on a large set of subpopulations derived from a synthetic SCM. The work establishes a practical bridge between causal inference and ML, releasing a synthetic dataset for $PNS$-bound evaluation and discussing data-collection strategies and limitations for real-world deployment.

Abstract

Probabilities of causation play a crucial role in modern decision-making. This paper addresses the challenge of predicting probabilities of causation for subpopulations with insufficient data using machine learning models. Tian and Pearl first defined and derived tight bounds for three fundamental probabilities of causation: the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of necessity (PN). However, estimating these probabilities requires both experimental and observational distributions specific to each subpopulation, which are often unavailable or impractical to obtain with limited population-level data. We assume that the probabilities of causation for each subpopulation are determined by its characteristics. To estimate these probabilities for subpopulations with insufficient data, we propose using machine learning models that draw insights from subpopulations with sufficient data. Our evaluation of multiple machine learning models indicates that, given sufficient population-level data and an appropriate choice of machine learning model and activation function, PNS can be effectively predicted. Through simulation studies, we show that our multilayer perceptron (MLP) model with the Mish activation function achieves a mean absolute error (MAE) of approximately 0.02 in predicting PNS for 32,768 subpopulations using data from around 2,000 subpopulations.

Estimating Probabilities of Causation with Machine Learning Models

TL;DR

Estimating probabilities of causation for subpopulations with limited data is challenging under Structural Causal Models. The authors propose a machine learning framework to predict the bounds of , , and by training on subpopulations with sufficient data, focusing on accurately predicting . Evaluating five models, they find that a Multilayer Perceptron with Mish activation delivers the best performance, achieving a mean absolute error near when predicting bounds on a large set of subpopulations derived from a synthetic SCM. The work establishes a practical bridge between causal inference and ML, releasing a synthetic dataset for -bound evaluation and discussing data-collection strategies and limitations for real-world deployment.

Abstract

Probabilities of causation play a crucial role in modern decision-making. This paper addresses the challenge of predicting probabilities of causation for subpopulations with insufficient data using machine learning models. Tian and Pearl first defined and derived tight bounds for three fundamental probabilities of causation: the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of necessity (PN). However, estimating these probabilities requires both experimental and observational distributions specific to each subpopulation, which are often unavailable or impractical to obtain with limited population-level data. We assume that the probabilities of causation for each subpopulation are determined by its characteristics. To estimate these probabilities for subpopulations with insufficient data, we propose using machine learning models that draw insights from subpopulations with sufficient data. Our evaluation of multiple machine learning models indicates that, given sufficient population-level data and an appropriate choice of machine learning model and activation function, PNS can be effectively predicted. Through simulation studies, we show that our multilayer perceptron (MLP) model with the Mish activation function achieves a mean absolute error (MAE) of approximately 0.02 in predicting PNS for 32,768 subpopulations using data from around 2,000 subpopulations.

Paper Structure

This paper contains 20 sections, 1 theorem, 18 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Structural Causal Model
  5. Data Generating Process
  6. Informer Data
  7. Sample Collection
  8. Data for Machine Learning Models
  9. Machine Learning Prediction
  10. Support Vector Machine
  11. Random Forest
  12. Gradient Boosting Decision Trees
  13. Transformer
  14. Multilayer Perceptron
  15. Experimental Comparison
  16. ...and 5 more sections

Key Result

Theorem 5

If $Y$ is monotonic relative to $X$, then PNS, PN, and PS are all identifiable, and

Figures (5)

  • Figure 1: Framework for Causal Data Generation and Machine Learning Prediction.
  • Figure 2: Comparison of true and predicted values across different models for both lower and upper bounds.
  • Figure 3: Confusion matrices of SVM, RF, GBDT, and Transformer models.
  • Figure 4: Confusion matrices of MLP with different activation functions: ReLU, Leaky ReLU, and Mish for both lower and upper bounds.
  • Figure 5: Comparison of MLP (Mish) for lower and upper bounds.

Theorems & Definitions (5)

  • Definition 1: Probability of necessity (PN)
  • Definition 2: Probability of sufficiency (PS)
  • Definition 3: Probability of necessity and sufficiency (PNS)
  • Definition 4
  • Theorem 5