  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.
