We show how to construct the implied copula process of response values from a Bayesian additive regression tree (BART) model with prior on the leaf node variances. This copula process, defined on the covariate space, can be combined with any marginal distribution for the dependent variable to construct a flexible distributional BART model. Bayesian inference is performed via Markov chain Monte Carlo on an augmented posterior, where we show that key sampling steps can be realized as those of Chipman et al. (2010), preserving scalability and computational efficiency despite the high dimensionality of the copula process. The posterior predictive distribution from the copula process model is derived in closed form as the push-forward of the posterior predictive distribution of the underlying BART model with an optimal transport map. Under suitable conditions, and allowing for model misspecification, we show that the model-implied conditional density closest in Kullback-Leibler divergence to the true data-generating process exists, is unique and has a closed form expression. We further establish that fractional posteriors for the conditional densities at the observed covariate values concentrate around this projection at a rate that adapts to the smoothness of the corresponding regression function and to the number of relevant covariates. Simulation studies demonstrate improved accuracy of distributional predictions compared to the original BART model and leading benchmarks. Applications to five real datasets with 506 to 515,345 observations and 8 to 90 covariates further highlight the efficacy and scalability of our proposed BART copula process model.

翻译：暂无翻译