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

TL;DR

This paper offers a straightforward, measure-theory-free derivation showing how the stick-breaking process directly leads to the Chinese restaurant process, clarifying their connection in Dirichlet process representations.

Contribution

It provides an elementary, direct proof of the relationship between the stick-breaking process and the CRP, simplifying understanding for nonexperts.

Findings

Direct derivation of CRP from stick-breaking process

Clarification of the natural emergence of the stick-breaking representation from CRP

Elimination of measure theory in the proof

Abstract

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.