  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.
