Adaptive Shrinkage in Pólya Tree Type Models

Abstract

We introduce a hierarchical generalization to the Pólya tree that incorporates locally adaptive shrinkage to data features of different scales, while maintaining analytical simplicity and computational efficiency. Inference under the new model proceeds efficiently using general recipes for conjugate hierarchical models, and can be completed extremely efficiently for data sets with large numbers of observations. We illustrate in density estimation that the achieved adaptive shrinkage results in proper smoothing and substantially improves inference. We evaluate the performance of the model through simulation under several schematic scenarios carefully designed to be representative of a variety of applications. We compare its performance to that of the Pólya tree, the optional Pólya tree, and the Dirichlet process mixture. We then apply our method to a flow cytometry data with 455,472 observations to achieve fast estimation of a large number of univariate and multivariate densities, and investigate the computational properties of our method in that context. In addition, we establish theoretical guarantees for the model including absolute continuity, full nonparametricity, and posterior consistency. All proofs are given in the Supplementary Material (Ma, 2016).