More aspects of Pólya tree distributions for statistical modelling. (English) Zbl 0820.62016

Summary: 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 P. Diaconis and D. Freedman [ibid. 14, 68-87 (1986; Zbl 0595.62023)] 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 D. A. Berry and R. Christensen [ibid. 7, 558-568 (1979; Zbl 0407.62018)] is reanalyzed with a Polya tree model.


62E10 Characterization and structure theory of statistical distributions
62A01 Foundations and philosophical topics in statistics
62G99 Nonparametric inference
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