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September, 1994 More Aspects of Polya Tree Distributions for Statistical Modelling
Michael Lavine
Ann. Statist. 22(3): 1161-1176 (September, 1994). DOI: 10.1214/aos/1176325623


The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information such as monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Christensen is reanalyzed with a Polya tree model.


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Michael Lavine. "More Aspects of Polya Tree Distributions for Statistical Modelling." Ann. Statist. 22 (3) 1161 - 1176, September, 1994.


Published: September, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0820.62016
MathSciNet: MR1311970
Digital Object Identifier: 10.1214/aos/1176325623

Primary: 62A15
Secondary: 62G07 , 62G99

Keywords: Dirichlet processes , Nonparametric regression , Robust Bayes , tail-free processes

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 3 • September, 1994
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