Quantile pyramids for Bayesian nonparametrics

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, there are two candidate likelihood functions, based on the need to work with a finite set of partitions. Both likelihood functions factorise in precisely the same way as for the quantile pyramid priors. While analytic summaries of posterior distributions are too complicated, updating with Markov chain Monte Carlo methods is quite straightforward. Among special cases of quantile pyramids we have the Dirichlet process. We give conditions securing the existence of an absolute continuous quantile process, and discuss consistency of the sequence of posterior distributions. Illustrations are included