The Annals of Statistics

Quantile pyramids for Bayesian nonparametrics

Nils Lid Hjort and Stephen G. Walker

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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.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 105-131.

First available in Project Euclid: 16 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 62F15: Bayesian inference

Consistency Dirichlet process nonparametric Bayes Bernshteĭn–von Mises theorem quantile pyramids random quantiles


Hjort, Nils Lid; Walker, Stephen G. Quantile pyramids for Bayesian nonparametrics. Ann. Statist. 37 (2009), no. 1, 105--131. doi:10.1214/07-AOS553.

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