  Prior distributions play a crucial role in Bayesian approaches to clustering.
Two commonly-used prior distributions are the Dirichlet and Pitman-Yor
processes. In this paper, we investigate the predictive probabilities that
underlie these processes, and the implicit "rich-get-richer" characteristic of
the resulting partitions. We explore an alternative prior for nonparametric
Bayesian clustering -- the uniform process -- for applications where the
"rich-get-richer" property is undesirable. We also explore the cost of this
process: partitions are no longer exchangeable with respect to the ordering of
variables. We present new asymptotic and simulation-based results for the
clustering characteristics of the uniform process and compare these with known
results for the Dirichlet and Pitman-Yor processes. We compare performance on a
real document clustering task, demonstrating the practical advantage of the
uniform process despite its lack of exchangeability over orderings.
