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


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:

Ismael Castillo

Sorbonne University ( email )

UFR 927, 4 Place Jussieu
Paris, F-75252

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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