Uncertainty Quantification for Bayesian CART

TL;DR

This paper develops a formal inferential framework for Bayesian CART, demonstrating adaptive confidence bands and optimal inference for smooth functionals, advancing uncertainty quantification in tree-based regression models.

Contribution

It introduces a new Bayesian CART prior based on a g-type structure that captures tree topology correlation, enabling adaptive and optimal uncertainty quantification.

Findings

Bayesian CART attains near rate-minimax posterior concentration.

Constructs adaptive confidence bands with uniform coverage.

Enables optimal inference for smooth functionals.

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.