Open Access
February 2009 Quantile pyramids for Bayesian nonparametrics
Nils Lid Hjort, Stephen G. Walker
Ann. Statist. 37(1): 105-131 (February 2009). DOI: 10.1214/07-AOS553


Pólya trees fix partitions and use random probabilities in order to construct random probability measures. With quantile pyramids we instead fix probabilities and use random partitions. For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, yet we show that the limiting version of the prior exists. We also discuss and investigate an alternative model based on the so-called substitute likelihood. Both approaches factorize in a convenient way leading to relatively straightforward analysis via MCMC, since analytic summaries of posterior distributions are too complicated. We give conditions securing the existence of an absolute continuous quantile process, and discuss consistency and approximate normality for the sequence of posterior distributions. Illustrations are included.


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Nils Lid Hjort. Stephen G. Walker. "Quantile pyramids for Bayesian nonparametrics." Ann. Statist. 37 (1) 105 - 131, February 2009.


Published: February 2009
First available in Project Euclid: 16 January 2009

zbMATH: 1360.62124
MathSciNet: MR2488346
Digital Object Identifier: 10.1214/07-AOS553

Primary: 60G35 , 62F15

Keywords: Bernshteĭn–von Mises theorem , consistency , Dirichlet process , nonparametric Bayes , quantile pyramids , random quantiles

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • February 2009
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