What is a good matching of probability measures? A counterfactual lens on transport maps
什么是概率测度的良好匹配？传输映射的反事实视角AI 解读 ↓
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什么是概率测度的良好匹配？传输映射的反事实视角
- 研究问题
- 当相同边际分布允许无穷多个确定性传输映射时，应如何理解不同单调匹配的差异，以及哪些因果假设能够支持特定反事实映射？
- 方法与思路
- 作者比较循环单调、保分位数和三角单调映射，推导其等价条件，并将结构因果模型中的反事实推理表述为固定边际间的映射选择。
- 材料陈述的贡献
- 摘要称作者建立三类传输映射等价的充要条件，并说明在何种因果图与结构方程条件下，反事实映射能够与经典统计传输相一致。
- 阅读关注点
- 核对三类单调性与等价条件的精确定义、因果图和结构方程各承担什么限制，以及仅有边际分布时无法检验的假设如何决定映射选择。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未列出具体等价条件或因果限制，不能据此把最优传输映射直接解释为已识别的反事实。
原文摘要
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.