Multiscale Analysis of Bayesian Cart

65 Pages Posted: 21 Oct 2019

See all articles by Ismael Castillo

Ismael Castillo

Sorbonne University

Veronika Rockova

University of Chicago - Booth School of Business

Date Written: October 1, 2019

Abstract

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

Castillo, Ismael and Rockova, Veronika, Multiscale Analysis of Bayesian Cart (October 1, 2019). University of Chicago, Becker Friedman Institute for Economics Working Paper No. 2019-127. Available at SSRN: https://ssrn.com/abstract=3472021

Ismael Castillo

Sorbonne University ( email )

UFR 927, 4 Place Jussieu
Paris, F-75252
France

Veronika Rockova (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

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