Doubly Robust Kernel Statistics for Testing Distributional Treatment Effects
This provides a tool for causal inference researchers to detect effects beyond the mean, though it is incremental as it builds on existing Counterfactual Mean Embeddings.
The paper tackles the problem of testing for distributional causal effects, where treatment affects higher-order moments and multidimensional outcomes, by proposing new doubly robust estimators for distributional embeddings in RKHS, leading to improved convergence rates and new permutation-based tests.
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.