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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 57 (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 a Global Sensitivity Measure for Bayesian Inference
"... We define a global sensitivity measure that is useful in assessing sensitivity to deviations from a specified prior. We argue that this measure has a common interpretation irrespective of the context of the problem, or the unit of measurements, and is therefore easy to interpret. We also study the a ..."
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We define a global sensitivity measure that is useful in assessing sensitivity to deviations from a specified prior. We argue that this measure has a common interpretation irrespective of the context of the problem, or the unit of measurements, and is therefore easy to interpret. We also study the asymptotic behavior of this global sensitivity measure. We find that it does not always converge to 0 as the sample size goes to infinity. We also show that, under certain conditions, this measure does go to 0 as the sample size goes to infinity. Thus, unlike the usual global sensitivity measure range, this measure behaves asymptotically like the usual local sensitivity measure. 1 AMS 1991 subject classifications. Primary 62F35; secondary 62C10 Key words and phrases. Bayesian robustness, global sensitivity, asymptotics. 1 1 Introduction In a Bayesian analysis involving a subjectively elicited prior, one is usually concerned with sensitivity to deviations from the specified prior, 0 . In ...