Prior Distributions for Partitions in Bayesian Nonparametrics

Prior distributions for unknown data distributions play an important role in nonparametric Bayesian statistics. A commonly-used prior distribution for an unknown data distribution is the Dirichlet process, which induces a random partition on the observations from the unknown data distribution. We investigate the prediction rule that underlies the Dirichlet process prior and the implicit "rich-get-richer" characteristics of random partitions generated by this process. To provide more flexibility for the modeling of random partitions, we present two alternative prior distributions for random partitions: the Pitman-Yor process and a uniform process. We present several asymptotic results for partitions under each process as well as a simulation-based evaluation of partition properties in small samples. We also discuss the exchangeability of partitions under each prediction rule. We give special focus to the uniform process which does not share the same "rich-get-richer" property as the Dirichlet process, which would be advantageous in applications where that implicit property is not reasonable.