Quantile pyramids for Bayesian nonparametrics

Details Quantile pyramids for Bayesian nonparametrics Quantile pyramids for Bayesian nonparametrics

2009-02-25

arxiv.org/abs/0902.4410v1

Annals of Statistics 2009, Vol. 37, No. 1, 105-131

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-AOS553 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/07-AOS553

P\'{o}lya 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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