In this paper we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph. Different graph structures lead to different specifications of the optimal transport problem. For instance, a fully connected graph yields standard optimal transport, a linear graph structure corresponds to adapted optimal transport, and an empty graph leads to a notion of optimal transport related to CO-OT, Gromov-Wasserstein distances and factored OT. We derive different characterizations of causal transport plans and introduce Wasserstein distances between causal models that respect the underlying graph structure. We show that average treatment effects are continuous with respect to causal Wasserstein distances and small perturbations of structural causal models lead to small deviations in causal Wasserstein distance. We also introduce an interpolation between causal models based on causal Wasserstein distance and compare it to standard Wasserstein interpolation.

翻译：本文针对有向图给定的因果结构，引入了一种最优输运的变体。不同的图结构对应最优输运问题的不同具体形式。例如：全连接图对应标准最优输运，线性图结构对应适应性最优输运，空图则导致一种与CO-OT、Gromov-Wasserstein距离及分解式最优输运相关的概念。我们推导了因果输运方案的不同刻画，并引入了尊重底层图结构的因果模型间的Wasserstein距离。研究表明，平均处理效应关于因果Wasserstein距离具有连续性，且结构因果模型的微小扰动仅导致因果Wasserstein距离的微小偏差。我们还基于因果Wasserstein距离引入了因果模型间的插值方法，并将其与标准Wasserstein插值进行了比较。