Veronika Rockova

UNCERTAINTY QUANTIFICATION FOR BAYESIAN CART

This work develops a formal inferential framework for Bayesian tree-based regression. We afford new insights into Bayesian CART in the context of structured wavelet shrinkage. We reframe Bayesian CART priors as g-type priors which depart from the typical wavelet product priors by harnessing correlation induced by the tree topology. We show that practically used Bayesian CART priors attain adaptive rate-minimax posterior concentration in the supremum norm in Gaussian white noise and non-parametric regression. For inference with certain functionals, we derive an adaptive non-parametric Bernstein-von Mises theorem implying that quantile credible sets are optimal confidence sets. For the fundamental goal of uncertainty quantification, we construct adaptive confidence bands with uniform coverage for the regression function under self-similarity.