  Distributional Random Forest (DRF) is a flexible forest-based method to
estimate the full conditional distribution of a multivariate output of interest
given input variables. In this article, we introduce a variable importance
algorithm for DRFs, based on the well-established drop and relearn principle
and MMD distance. While traditional importance measures only detect variables
with an influence on the output mean, our algorithm detects variables impacting
the output distribution more generally. We show that the introduced importance
measure is consistent, exhibits high empirical performance on both real and
simulated data, and outperforms competitors. In particular, our algorithm is
highly efficient to select variables through recursive feature elimination, and
can therefore provide small sets of variables to build accurate estimates of
conditional output distributions.
