A new proof of the stick-breaking representation of Dirichlet processes

Abstract

The stick-breaking representation is one of the fundamental properties of the Dirichlet process. It represents the random probability measure as a discrete random sum whose weights and atoms are formed by independent and identically distributed sequences of beta variates and draws from the normalized base measure of the Dirichlet process parameter. It is used extensively in posterior simulation for statistical models with Dirichlet processes. The original proof of Sethuraman (Stat Sin 4:639–650, 1994) relies on an indirect distributional equation and does not encourage an intuitive understanding of the property. In this paper, we give a new proof of the stick-breaking representation of the Dirichlet process that provides an intuitive understanding of the theorem. The proof is based on the posterior distribution and self-similarity properties of the Dirichlet process.