A simple proof of Pitman-Yor's Chinese restaurant process from its stick-breaking representation

Abstract

For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The Pitman--Yor process provides a simple and mathematically tractable generalization, allowing for a very flexible control of the clustering behaviour. Two commonly used representations of the Pitman--Yor process are the stick-breaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. However, the usual proof of the connection between them is indirect and involves measure theory. We provide here an elementary proof of Pitman--Yor's Chinese Restaurant process from its stick-breaking representation.

arXiv comment11 pages

References in corpus (8)

• Power-law distributions in empirical data
• Are Gibbs-type priors the most natural generalization of the Dirichlet process?
• Bayesian semiparametric inference for multivariate doubly-interval-censored data
• Inconsistency of Pitman-Yor process mixtures for the number of components
• Model-based clustering using copulas with applications
• Stochastic approximations to the Pitman-Yor process
• Bayesian nonparametric inference for discovery probabilities: credible intervals and large sample asymptotics
• Some Diffusion Processes Associated With Two Parameter Poisson-Dirichlet Distribution and Dirichlet Process