Applications of the Two-Parameter Poisson-Dirichlet Distribution
Advisor: Thomas Ferguson
This dissertation investigates properties and applications of the two-parameter Poisson-Dirichlet distribution. To begin, we establish notation and describe the background history of this distribution. It has been shown that certain previously-known distributions for probability sequences are marginal cases of this two-parameter distribution. Moreover, this distribution is in some sense maximal among those distributions which exhibit invariance under size-biased permutation. We then introduce the related Poisson-Dirichlet process and shows some of its more basic properties. This generalizes the Dirichlet process, which has been a popular tool for Bayesian statisticians since its discovery. In particular, we investigate at length a new special case of the Poisson-Dirichlet process. Therein, we uncover a new class of densities on the simplex and discuss computation of its moments. We also consider various realizations of the posterior distribution of the Poisson-Dirichlet process. We then change perspectives. A one-parameter Poisson-Dirichlet distribution has been used in biological contexts for some time, and we consider the use of the two-parameter version in this framework. This leads to a generalization of the Ewens Sampling Formula. Conversely, we find a wealth of mathematical results which we would not have uncovered absent this biological perspective. These include a general moments formula for the Poisson-Dirichlet process and an extensive look at the behavior and growth of the number of species in this biological context. These results lead to several methods for estimating the parameters of the Poisson-Dirichlet distribution. In addition to the traditional maximum likelihood estimates, we give useful alternatives based on both the aforementioned biological work and separate research by Pitman et al., who take a totally different approach to the Poisson-Dirichlet distribution.