  We present new insights into causal inference in the context of Heterogeneous
Treatment Effects by proposing natural variants of Random Forests to estimate
the key conditional distributions. To achieve this, we recast Breiman's
original splitting criterion in terms of Wasserstein distances between
empirical measures. This reformulation indicates that Random Forests are well
adapted to estimate conditional distributions and provides a natural extension
of the algorithm to multivariate outputs. Following the philosophy of Breiman's
construction, we propose some variants of the splitting rule that are
well-suited to the conditional distribution estimation problem. Some
preliminary theoretical connections are established along with various
numerical experiments, which show how our approach may help to conduct more
transparent causal inference in complex situations.
