Departmental Colloquia: Veronika Rockova

VERONIKA ROCKOVA 

 

 

Department of Econometrics and Statistics
University of Chicago Booth School of Business

 

 

On Theory for BART

 

ABSTRACT

Since their inception in the 1980¹s, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal nodes of recursive partitioning. Trees are powerful, yet susceptible to over-fitting. Strategies against overfitting have traditionally relied on pruning greedily grown trees. The Bayesian framework offers an alternative remedy against overfitting through priors. Roughly  speaking, a good prior charges smaller trees where overfitting does not occur. While the consistency of random histograms, trees and their ensembles has been studied quite extensively, the theoretical understanding of the Bayesian counterparts has been missing. In this paper, we take first steps towards understanding why/when do Bayesian trees and forests not overfit. To address this question, we study the speed at which the posterior concentrates around the true smooth regression function. We propose a spike-and-tree variant of the popular Bayesian CART prior and establish new theoretical results showing that regression trees (and forests) (a) are capable of recovering smooth regression surfaces (with smoothness not exceeding one), achieving optimal rates up to a log factor, (b) can adapt to the unknown level of smoothness and (c) can perform effective dimension reduction when p > n. Going further, we also show semi and non-parametric Bernstein-von Mises theorems showing that BART is fundamentally justified from a frequentist point of view. These results provide a piece of missing theoretical evidence explaining why Bayesian trees (and additive variants thereof) have worked so well in practice.

 

 

Friday, 3/1/2019, 11:30 AM, BLOC 113