The Poisson–Dirichlet Distribution

Abstract

The focus of this chapter is the Poisson–Dirichlet distribution, the central topic of this book. We introduce this distribution and discuss various models that give rise to it. Following Kingman (J. Roy. Statist. Soc. B 37:1–22, 1975), the distribution is constructed through the gamma process. An alternative construction in (R. Arratia, A.D. Barbour and S. Tavaré in Combin. Probab. Comput. 8:407–416 1999) is also included, where a scale-invariant Poisson process is used. The density functions of the marginal distributions are derived through Perman’s formula. Closely related topics such as the GEM distribution, the Ewens sampling formula, and the Dirichlet process are investigated in detail through the study of urn models. The required terminology and properties of Poisson processes and Poisson random measures can be found in Appendix A.