Hierarchical priors for Bayesian CART shrinkage

The Bayesian CART (classiication and regression tree) approach proposed by Chipman, George and McCulloch (1998) entails putting a prior distribution on the set of all CART models and then using stochastic search to select a model. The main thrust of this paper is to propose a new class of hierarchical priors which enhance the potential of this Bayesian approach. These priors indicate a preference for smooth local mean structure, resulting in tree models which shrink predictions from adjacent terminal node towards each other. Past methods for tree shrinkage have searched for trees without shrinking, and applied shrinkage to the identiied tree only after the search. By using hierarchical priors in the stochastic search, the proposed method searches for shrunk trees that t well and afterwards improves the tree through shrinkage of predictions.