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Hjort, Nils Lid; Walker, Stephen G.

Quantile pyramids for Bayesian nonparametrics. (English) Zbl 1360.62124

Ann. Stat. 37, No. 1, 105-131 (2009).
Summary: 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.

Cited in 19 Documents

MSC:

62G05 Nonparametric estimation
62F15 Bayesian inference
62G20 Asymptotic properties of nonparametric inference
60G35 Signal detection and filtering (aspects of stochastic processes)
60G57 Random measures
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References:

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