  We present a Bayesian nonparametric model for conditional distribution
estimation using Bayesian additive regression trees (BART). The generative
model we use is based on rejection sampling from a base model. Typical of BART
models, our model is flexible, has a default prior specification, and is
computationally convenient. To address the distinguished role of the response
in the BART model we propose, we further introduce an approach to targeted
smoothing which is possibly of independent interest for BART models. We study
the proposed model theoretically and provide sufficient conditions for the
posterior distribution to concentrate at close to the minimax optimal rate
adaptively over smoothness classes in the high-dimensional regime in which many
predictors are irrelevant. To fit our model we propose a data augmentation
algorithm which allows for existing BART samplers to be extended with minimal
effort. We illustrate the performance of our methodology on simulated data and
use it to study the relationship between education and body mass index using
data from the medical expenditure panel survey (MEPS).
