We propose to analyse the conditional distributional treatment effect (CoDiTE), which, in contrast to the more common conditional average treatment effect (CATE), is designed to encode a treatment's distributional aspects beyond the mean. We first introduce a formal definition of the CoDiTE associated with a distance function between probability measures. Then we discuss the CoDiTE associated with the maximum mean discrepancy via kernel conditional mean embeddings, which, coupled with a hypothesis test, tells us whether there is any conditional distributional effect of the treatment. Finally, we investigate what kind of conditional distributional effect the treatment has, both in an exploratory manner via the conditional witness function, and in a quantitative manner via U-statistic regression, generalising the CATE to higher-order moments. Experiments on synthetic, semi-synthetic and real datasets demonstrate the merits of our approach.

We present a new operator-free, measure-theoretic definition of the conditional mean embedding as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of marginal distributions has been defined rigorously, the existing operator-based approach of the conditional version lacks a rigorous definition, and depends on strong assumptions that hinder its analysis. Our definition does not impose any of the assumptions that the operator-based counterpart requires. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough analysis of its properties, including universal consistency. As natural by-products, we obtain the conditional analogues of the Maximum Mean Discrepancy and Hilbert-Schmidt Independence Criterion, and demonstrate their behaviour via simulations.