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We introduce a hierarchical nonparametric model for probability measuresbased on a multi-resolution transformation of probability distributions. Themodel allows a varying amount of shrinkage to be applied to data features ofdifferent scales and/or at different locations in the sample space, and thevarying shrinkage level is locally adaptive to the empirical behavior of thedata. Moreover, the model's hierarchical design---through a latent Markov treestructure---allows borrowing of information across locations and scales insetting the adaptive shrinkage level. Inference under the model proceedsefficiently using general recipes for conjugate hierarchical models. Weillustrate the work of the model in density estimation and evaluate itsperformance through simulation under several schematic scenarios carefullydesigned to be representative of a variety of applications. We compare itsperformance to those of several state-of-the-art nonparametric models---theP\'olya tree, the optional P\'olya tree, and the Dirichlet process mixture ofnormals. In addition, we establish several important theoretical properties forthe model including absolute continuity, full nonparametricity, and posteriorconsistency.

Markov adaptive P\'olya trees and multi-resolution adaptive shrinkage in nonparametric modeling