The Origins of the Stick Breaking Construction

Abstract

The origins of the stick breaking construction of Sethuraman (Statistica Sinica 4 639–650. 1994) are presented here. We show the equivalence of nonparametric priors and exchangeable distributions which automatically gives the posterior distribution of nonparametric priors. The Pólya exchangeable sequence of Blackwell and MacQueen (Ann. Statist. 1 353–355. 1973) corresponds to the Dirichlet process prior which again immediately gives the posterior distribution of of the Dirichlet prior. Studying the Pólya exchangeable sequence some more shows that the random probability measure distributed as a Dirichlet process is a random discrete probability measure and has something like a stick breaking construction. Under the condition that the parameter $\beta$ of the Dirichlet process is nonatomic, we show how to obtain the stick breaking representation of Sethuraman (Statistica Sinica 4 639–650. 1994). Noticing that this statement of the stick breaking construction puts no conditions $\beta$, Sethuraman (Statistica Sinica 4 639–650. 1994) used a different and a simpler method to obtain the stick breaking construction in full generality. Finally we present an example to remove a generally held misconception of Sethuraman (Statistica Sinica 4 639–650. 1994).