  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.
