In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. 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 causal optimal transport between the distributions of two discrete-time stochastic processes, 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 $G$-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 $G$-causal Wasserstein distances and small perturbations of structural causal models lead to small deviations in $G$-causal Wasserstein distance. We also introduce an interpolation between causal models based on $G$-causal Wasserstein distance and compare it to standard Wasserstein interpolation.

翻译：本文提出了一种适用于有向图$G$所定义因果结构的最优传输变体。不同的图结构会导出不同形式的最优传输问题。例如，全连接图对应经典最优传输，线性图结构对应于两个离散时间随机过程分布间的因果最优传输，而空图则导出一类与CO-OT、Gromov-Wasserstein距离及分解式OT相关的最优传输概念。我们推导了$G$-因果传输方案的不同表征形式，并建立了尊重底层图结构的因果模型间Wasserstein距离。我们证明平均处理效应关于$G$-因果Wasserstein距离具有连续性，且结构因果模型的微小扰动会导致$G$-因果Wasserstein距离的微小偏差。同时，我们提出基于$G$-因果Wasserstein距离的因果模型插值方法，并将其与标准Wasserstein插值进行对比。