Quantile pyramids for Bayesian nonparametrics

TL;DR

This paper introduces quantile pyramids, a novel Bayesian nonparametric approach that fixes probabilities and uses random partitions to model piecewise linear quantile functions, with theoretical guarantees and practical MCMC methods.

Contribution

It proposes a new prior based on fixing probabilities and random partitions, supporting piecewise linear quantile functions, and analyzes its properties and computational methods.

Findings

Existence of a limiting prior for quantile pyramids

Conditions for absolute continuity of the quantile process

Posterior distributions exhibit consistency and approximate normality

Abstract

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.