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:在本文中,我们介绍了一种适用于由基础有向图给出的因果结构的最优输运的变种。不同的图结构导致了最优输运问题的不同规范。例如,完全连接的图形导致标准的最优输运,线性图结构对应于适应性最优输运,而空图导致关于CO-OT,Gromov-Wasserstein距离和factored OT的最优输运的概念。我们推导了因果传输计划的不同特征,并介绍了尊重基础图结构的因果模型之间的Wasserstein距离。我们证明了平均治疗效应与因果Wasserstein距离连续,并且结构因果模型的小扰动导致因果Wasserstein距离小偏差。我们还介绍了基于因果Wasserstein距离的因果模型之间的插值,并将其与标准Wasserstein插值进行比较。