Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions

2026-02-21

AI Summary

This study addresses causal inference with discrete treatments and ordered discrete outcomes by investigating the bounds and identification of joint probabilities involving potential outcomes and observed variables, as well as their linear combinations. By introducing a novel class of monotonicity assumptions, the problem of bounding joint probabilities is transformed into a linear programming formulation. A stronger monotonicity condition is further proposed to achieve point identification. Integrating the potential outcomes framework with discrete variable modeling techniques, the approach ensures tight and exact identification both theoretically and algorithmically. Numerical simulations and real-data experiments demonstrate the method’s effectiveness, substantially expanding the identifiable range and improving the estimation accuracy of discrete causal effects.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.

Problem

• causal inference
• joint probabilities
• potential outcomes
• monotonicity
• identification

Innovation

• monotonicity assumptions
• joint probabilities of potential outcomes
• causal inference
• linear programming
• identification