A simple proof of Pitman-Yors Chinese restaurant process from its stick-breaking representation

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--Yors Chinese Restaurant process from its stick-breaking representation.