Density estimation trees

In this paper we develop density estimation trees (DETs), the natural analog of classification trees and regression trees, for the task of density estimation. We consider the estimation of a joint probability density function of a d-dimensional random vector X and define a piecewise constant estimator structured as a decision tree. The integrated squared error is minimized to learn the tree. We show that the method is nonparametric: under standard conditions of nonparametric density estimation, DETs are shown to be asymptotically consistent. In addition, being decision trees, DETs perform automatic feature selection. They empirically exhibit the interpretability, adaptability and feature selection properties of supervised decision trees while incurring slight loss in accuracy over other nonparametric density estimators. Hence they might be able to avoid the curse of dimensionality if the true density is sparse in dimensions. We believe that density estimation trees provide a new tool for exploratory data analysis with unique capabilities.