Kullback-Leibler Consistency of $p$-dimensional Pólya Tree Posteriors and Differential Entropy Estimation

We exploit the multiplicative structure of Pólya 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 Pólya Tree posteriors. These results motivate a novel differential entropy estimator that is consistent under mild conditions on large dimensions.