Doubly Robust Kernel Statistics for Testing Distributional Treatment Effects

TL;DR

This paper introduces doubly robust kernel-based estimators for testing distributional causal effects, capable of handling complex outcomes and higher moments, with proven robustness and improved convergence.

Contribution

It develops new doubly robust estimators within RKHS for distributional causal effects, enhancing existing methods with better convergence and validity.

Findings

Proposed estimators are doubly robust and have improved convergence rates.

New permutation tests effectively detect distributional causal effects.

Theoretical and experimental validation confirms test validity.

Abstract

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.