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SUMMARY:Posterior concentration for Bayesian regression trees and their en
 sembles - Stephanie van der Pas\, University of Leiden
DTSTART:20181123T160000Z
DTEND:20181123T170000Z
UID:TALK109678@talks.cam.ac.uk
CONTACT:Dr Sergio Bacallado
DESCRIPTION:Since their inception in the 1980's\, regression trees have be
 en one of the more widely used nonparametric prediction methods. Tree-stru
 ctured methods yield a histogram reconstruction of the regression surface\
 , where the bins correspond to terminal nodes of recursive partitioning. T
 rees are powerful\, yet susceptible to overfitting. Strategies against ove
 rfitting have traditionally relied on pruning greedily grown trees. The Ba
 yesian framework offers an alternative remedy against overfitting through 
 priors. Roughly speaking\, a good prior charges smaller trees where overfi
 tting does not occur. In this paper\, we take a step towards understanding
  why/when do Bayesian trees and their ensembles not overfit. We study the 
 speed at which the posterior concentrates around the true smooth regressio
 n function. We propose a spike-and-tree variant of the popular Bayesian CA
 RT prior and establish new theoretical results showing that regression tre
 es (and their ensembles) a) are capable of recovering smooth regression su
 rfaces\, achieving optimal rates up to a log factor\, b) can adapt to the 
 unknown level of smoothness and c) can perform effective dimension reducti
 on. These results provide a piece of missing theoretical evidence explaini
 ng why Bayesian trees (and additive variants thereof) have worked so well 
 in practice.
LOCATION:MR12
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