An elementary derivation of the Chinese restaurant process from Sethuraman's stick-breaking process

The Chinese restaurant process (CRP) and the stick-breaking process are the two most commonly used representations of the Dirichlet process. However, the usual proof of the connection between them is indirect, relying on abstract properties of the Dirichlet process that are difficult for nonexperts to verify. This short note provides a direct proof that the stick-breaking process leads to the CRP, without using any measure theory. We also discuss how the stick-breaking representation arises naturally from the CRP.