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More Aspects of Polya Tree Distributions for Statistical Modelling
 Ann. Statist
, 1994
"... : 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 neighborhoo ..."
Abstract

Cited by 93 (1 self)
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: 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 like 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 Chris...
On Posterior Consistency in Selection Models
, 1999
"... Selection models are appropriate when the probability that a potential datum enters the sample is a nondecreasing function of the numeric value of the datum. It is rarely justifiable to model this function, called the weight function, with a specific parametric form, but is appealing to model with a ..."
Abstract

Cited by 1 (0 self)
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Selection models are appropriate when the probability that a potential datum enters the sample is a nondecreasing function of the numeric value of the datum. It is rarely justifiable to model this function, called the weight function, with a specific parametric form, but is appealing to model with a nonparametric prior centered around a parametric form. The Bayesian analysis with Dirichlet process prior for the weight function is considered and it is proved that the posterior is consistent under the weak topology.