What is a good matching of probability measures? A counterfactual lens on transport maps

2025-09-19

AI Summary

Existing optimal transport approaches to causal inference suffer from non-uniqueness of transport maps and conceptual conflation among multivariate monotonicity notions, leaving the selection of counterfactual transport maps under fixed marginals theoretically unjustified. Method: We establish necessary and sufficient conditions for the equivalence of cyclically monotone, quantile-preserving, and triangularly monotone maps; formulate counterfactual inference as a transport map selection problem under fixed marginals; and systematically characterize map identifiability under causal graphs and structural equation models. Contribution/Results: Our work unifies statistical optimal transport and causal inference within a single theoretical framework. It precisely delineates the applicability domains of each map class and reveals how causal assumptions fundamentally constrain admissible map structures. By grounding counterfactual reasoning in testable transport-theoretic principles, the study provides a rigorous, empirically verifiable foundation for marginal-preserving causal mapping—resolving long-standing ambiguities in transport-based counterfactual estimation.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Coupling probability measures lies at the core of many problems in statistics and machine learning, from domain adaptation to transfer learning and causal inference. Yet, even when restricted to deterministic transports, such couplings are not identifiable: two atomless marginals admit infinitely many transport maps. The common recourse to optimal transport, motivated by cost minimization and cyclical monotonicity, obscures the fact that several distinct notions of multivariate monotone matchings coexist. In this work, we first carry a comparative analysis of three constructions of transport maps: cyclically monotone, quantile-preserving and triangular monotone maps. We establish necessary and sufficient conditions for their equivalence, thereby clarifying their respective structural properties. In parallel, we formulate counterfactual reasoning within the framework of structural causal models as a problem of selecting transport maps between fixed marginals, which makes explicit the role of untestable assumptions in counterfactual reasoning. Then, we are able to connect these two perspectives by identifying conditions on causal graphs and structural equations under which counterfactual maps coincide with classical statistical transports. In this way, we delineate the circumstances in which causal assumptions support the use of a specific structure of transport map. Taken together, our results aim to enrich the theoretical understanding of families of transport maps and to clarify their possible causal interpretations. We hope this work contributes to establishing new bridges between statistical transport and causal inference.

Problem

• Compares multivariate monotone transport maps for probability couplings
• Links counterfactual reasoning to transport map selection in causal models
• Identifies conditions where causal and statistical transports coincide

Innovation

• Comparative analysis of multivariate monotone transport maps
• Connecting counterfactual reasoning with transport map selection
• Identifying causal conditions for statistical transport equivalence