Dirichlet Process, Ewens Sampling Formula, and Chinese Restaurant Process

Abstract

The Dirichlet process is a random probability measure and its realization is discrete almost surely. Therefore, there may be duplications among a sample from a distribution having the Dirichlet process. The distribution of this duplication is well known as the Ewens sampling formula. This formula is derived by another model, for example, the Chinese restaurant process. The Ewens sampling formula is related with the Donnelly–Tavare–Griffiths formula I and II, the GEM distribution, and the Poisson–Dirichlet distribution. The Donnelly–Tavare–Griffiths formula II is related with the Yule distribution and the Waring distribution. The distribution of the number of distinct components of the Ewens sampling formula asymptotically converges to normal distribution. It converges also to the shifted Poisson distribution under the condition resembled to that of Poisson law of small number. As a formula related to the Ewens sampling formula, the Pitman sampling formula is well known. It is also derived by the Chinese restaurant process.