J E Paguyo (McMaster University)

Title: Asymptotic behavior of the hierarchical Pitman-Yor process.

Abstract: The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base measure through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero gives the hierarchical Dirichlet process (HDP), first introduced by Teh et al. in the machine learning literature.

In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. In the first part, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. In the second part, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. This talk is based on joint work with Stefano Favaro and Shui Feng.