Multiscale Analysis of Bayesian Cart
65 Pages Posted: 21 Oct 2019
Date Written: October 1, 2019
This paper affords new insights about Bayesian CART in the context of structured wavelet shrinkage. We show that practically used Bayesian CART priors lead to adaptive rate-minimax posterior concentration in the supremum norm in Gaussian white noise, performing optimally up to a logarithmic factor. To further explore the benefits of structured shrinkage, we propose the g-prior for trees, which departs from the typical wavelet product priors by harnessing correlation induced by the tree topology. Building on supremum norm adaptation, an adaptive non-parametric Bernstein–von Mises theorem for Bayesian CART is derived using multi- scale techniques. For the fundamental goal of uncertainty quantification, we construct adaptive confidence bands with uniform coverage for the regression function under self-similarity.
Keywords: Bayesian CART, Posterior Concentration, Non-parametric Bernstein–von Mises theorem, Recursive Partitioning, Regression Trees
JEL Classification: 62G20, 62G15
Suggested Citation: Suggested Citation