Kullback-Leibler Consistency of $p$-dimensional P\'olya Tree Posteriors and Differential Entropy Estimation

Abstract

We exploit the multiplicative structure of P\'olya Tree priors to establish novel consistency results on $p$-dimensional trees, conditions to obtain Kullback-Leibler minimax contraction rates for univariate density estimation and a representation theorem of entropy functionals of P\'olya Tree posteriors. These results motivate a novel differential entropy estimator that is consistent under mild conditions on large dimensions.