Abstract
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 self-similarity. In addition, we show that tree-posteriors enable optimal inference in the form of efficient confidence sets for smooth functionals of the regression function.
Funding Statement
The first author gratefully acknowledges support from the Institut Universitaire de France and from the ANR Grant ANR-17-CE40-0001 (BASICS).
The second author gratefully acknowledges support from the James S. Kemper Foundation Faculty Research Fund at the University of Chicago Booth School of Business and the National Science Foundation (Grant DMS-1944740).
Funding Statement
The first author gratefully acknowledges support from the Institut Universitaire de France and from the ANR Grant ANR-17-CE40-0001 (BASICS).
The second author gratefully acknowledges support from the James S. Kemper Foundation Faculty Research Fund at the University of Chicago Booth School of Business and the National Science Foundation (Grant DMS-1944740).
Citation
Ismaël Castillo. Veronika Ročková. "Uncertainty quantification for Bayesian CART." Ann. Statist. 49 (6) 3482 - 3509, December 2021. https://doi.org/10.1214/21-AOS2093
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