Log In Sign Up

Multiscale Analysis of Bayesian CART

by   Ismaël Castillo, et al.

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 multiscale techniques. For the fundamental goal of uncertainty quantification, we construct adaptive confidence bands with uniform coverage for the regression function under self-similarity.


page 1

page 2

page 3

page 4


Ideal Bayesian Spatial Adaptation

Many real-life applications involve estimation of curves that exhibit co...

Spike and Slab Pólya tree posterior distributions

In the density estimation model, the question of adaptive inference usin...

Bayesian Wavelet Shrinkage with Beta Priors

We present a Bayesian approach for wavelet shrinkage in the context of n...

Gamma-Minimax Wavelet Shrinkage with Three-Point Priors

In this paper we propose a method for wavelet denoising of signals conta...

On Semi-parametric Bernstein-von Mises Theorems for BART

Few methods in Bayesian non-parametric statistics/ machine learning have...

Bayesian Multiscale Analysis of the Cox Model

Piecewise constant priors are routinely used in the Bayesian Cox proport...

The Bayes Lepski's Method and Credible Bands through Volume of Tubular Neighborhoods

For a general class of priors based on random series basis expansion, we...