# Quantile pyramids for Bayesian nonparametrics

Hjort, Nils Lid, Walker, Stephen G. (2009) Quantile pyramids for Bayesian nonparametrics. Annals of Statistics, 37 (1). pp. 105-131. ISSN 0090-5364. (doi:10.1214/07-AOS553) (KAR id:12613)

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## Abstract

Polya 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.

Item Type: Article 10.1214/07-AOS553 Consistency; Dirichlet process; nonparametric Bayes; Bernshtein-von Mises theorem; quantile pyramids; random quantiles Q Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Statistics Judith Broom 27 Mar 2009 18:45 UTC 28 May 2019 13:49 UTC https://kar.kent.ac.uk/id/eprint/12613 (The current URI for this page, for reference purposes)
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