With the widespread application of causal inference, it is increasingly important to have tools which can test for the presence of causal effects in a diverse array of circumstances. In this vein we focus on the problem of testing for \emph{distributional} causal effects, where the treatment affects not just the mean, but also higher order moments of the distribution, as well as multidimensional or structured outcomes. We build upon a previously introduced framework, Counterfactual Mean Embeddings, for representing causal distributions within Reproducing Kernel Hilbert Spaces (RKHS) by proposing new, improved, estimators for the distributional embeddings. These improved estimators are inspired by doubly robust estimators of the causal mean, using a similar form within the kernel space. We analyse these estimators, proving they retain the doubly robust property and have improved convergence rates compared to the original estimators. This leads to new permutation based tests for distributional causal effects, using the estimators we propose as tests statistics. We experimentally and theoretically demonstrate the validity of our tests.

翻译：随着因果推断的广泛应用，能够在多种场景下检验因果效应存在性的工具日益重要。针对这一问题，我们聚焦于检验"分布性"因果效应——即处理不仅影响均值，还影响分布的高阶矩以及多维或结构化结果的场景。我们基于先前提出的反事实均值嵌入框架，在再生核希尔伯特空间中表征因果分布，并提出了分布性嵌入的新改进估计量。这些改进估计量受因果均值的双重鲁棒估计量启发，在核空间中采用了类似形式。我们对这些估计量进行了分析，证明其保留了双重鲁棒性质，且相比原始估计量具有更快的收敛速度。基于此，我们利用所提出的估计量作为检验统计量，构建了新的置换检验方法用于检验分布性因果效应。理论分析与实验均验证了该检验的有效性。