The Annals of Statistics

More Aspects of Polya Tree Distributions for Statistical Modelling

Michael Lavine

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Abstract

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.

Article information

Source
Ann. Statist. Volume 22, Number 3 (1994), 1161-1176.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176325623

Digital Object Identifier
doi:10.1214/aos/1176325623

Mathematical Reviews number (MathSciNet)
MR1311970

Zentralblatt MATH identifier
0820.62016

JSTOR
links.jstor.org

Subjects
Primary: 62A15
Secondary: 62G07: Density estimation 62G99: None of the above, but in this section

Keywords
Dirichlet processes nonparametric regression robust Bayes tail-free processes

Citation

Lavine, Michael. More Aspects of Polya Tree Distributions for Statistical Modelling. Ann. Statist. 22 (1994), no. 3, 1161--1176. doi:10.1214/aos/1176325623. http://projecteuclid.org/euclid.aos/1176325623.


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