Posterior Concentration Rates for Bayesian Density Trees and Forest

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Density estimation is a useful statistical tool for sketching variations of data, profiling information content, and making risk optimal decisions. It consequently plays a fundamental role in a wide spectrum of statistical analyses and applications, such as two-sample comparison, data compression, and nonparametric evaluation of disease risk. In this work, we focus on tree based methods for density estimation under the Bayesian framework. First we show that the Bayesian density tree can achieve minimax convergence over the anisotropic Besov class, which implies that the method can adapt to spatially inhomogeneous features of the underlying density function, and can achieve fast convergence as the dimension increases. We also introduce a novel Bayesian model for density forests, and show that over the anisotropic H{\"o}lder space, forests can achieve faster convergence than trees, in an adaptive way. The Bayesian framework naturally endows a stochastic search scheme over the tree or forest space. For both Bayesian density trees and forests, we provide numerical results to illustrate their performance in the high-dimensional case.