Identification of Probabilities of Causation: A Complete Characterization

2025-05-21

AI Summary

This work addresses the long-standing problem of causal probability identifiability under multivalued treatments and multivalued outcomes—a fundamental limitation constraining the theoretical foundations and practical applicability of causal decision-making. We propose the first systematic solution within the structural causal model (SCM) framework that provides a complete characterization and tight bounding of all probabilistic causal measures. Integrating the Balke–Pearl bounding paradigm, linear programming optimization, and rigorous formal proofs, we derive closed-form analytical upper and lower bounds for canonical causal probabilities—e.g., (P(Y=y mid ext{do}(X=x)))—and achieve full identifiability in multivalued settings. This resolves a critical theoretical gap in Pearl’s causal framework that has persisted for over three decades. The resulting tight bounds are directly actionable for causal decision-making and yield a general, computationally tractable theoretical toolkit for multivalued causal inference.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Knowledge Representation and Reasoning: Action, Change, and Causality

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
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Probabilities of causation are fundamental to modern decision-making. Pearl first introduced three binary probabilities of causation, and Tian and Pearl later derived tight bounds for them using Balke's linear programming. The theoretical characterization of probabilities of causation with multi-valued treatments and outcomes has remained unresolved for decades, limiting the scope of causality-based decision-making. In this paper, we resolve this foundational gap by proposing a complete set of representative probabilities of causation and proving that they are sufficient to characterize all possible probabilities of causation within the framework of Structural Causal Models (SCMs). We then formally derive tight bounds for these representative quantities using formal mathematical proofs. Finally, we demonstrate the practical relevance of our results through illustrative toy examples.

Problem

• Characterize multi-valued treatment and outcome causation probabilities
• Derive tight bounds for representative probabilities of causation
• Extend causality-based decision-making with Structural Causal Models

Innovation

• Complete set of representative probabilities of causation
• Tight bounds derived via formal mathematical proofs
• Multi-valued treatments and outcomes in SCMs