Open Access
September 2017 Adaptive Shrinkage in Pólya Tree Type Models
Li Ma
Bayesian Anal. 12(3): 779-805 (September 2017). DOI: 10.1214/16-BA1021


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).


Download Citation

Li Ma. "Adaptive Shrinkage in Pólya Tree Type Models." Bayesian Anal. 12 (3) 779 - 805, September 2017.


Published: September 2017
First available in Project Euclid: 7 September 2016

zbMATH: 1384.62092
MathSciNet: MR3655876
Digital Object Identifier: 10.1214/16-BA1021

Primary: 62F15 , 62G99
Secondary: 62G07

Keywords: Bayesian nonparametrics , Density estimation , hierarchical models , multi-scale modeling

Vol.12 • No. 3 • September 2017
Back to Top