UNCERTAINTY QUANTIFICATION FOR BAYESIAN CART

• 成果类型: Article
• 署名作者: Castillo, Ismael; Rockova, Veronika
• 署名单位: Universite Paris Cite; Sorbonne Universite; Institut Universitaire de France; University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2093
• 发表日期: 2021
• 页码: 3482-3509
• 关键词: von mises theorems; polya tree; Nonparametric Regression; posterior distributions; l-p; contraction; rates; CLASSIFICATION; selection; METRICS

摘要

This work affords new insights into Bayesian CART in the context of structured wavelet shrinkage. The main thrust is to develop a formal inferential framework for Bayesian tree-based regression. We reframe Bayesian CART as a g-type prior which departs from the typical wavelet product priors by harnessing correlation induced by the tree topology. The practically used Bayesian CART priors are shown to attain adaptive near rate-minimax posterior concentration in the supremum norm in regression models. For the fundamental goal of uncertainty quantification, we construct adaptive confidence bands for the regression function with uniform coverage under selfsimilarity. In addition, we show that tree-posteriors enable optimal inference in the form of efficient confidence sets for smooth functionals of the regression function.