Functional Central Limit Theorems for Stick-Breaking Priors

Yaozhong Hu,Junxi Zhang

doi:10.1214/21-ba1290

Abstract

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stick-breaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical.

Highlights

Ever since the work of Ferguson (1973) the Dirichlet process has become a critical tool in Bayesian nonparametric statistics and has found applications in various areas, including machine learning, biological science, social science and so on
We are mainly concerned with three types of the asymptotics for a number of processes, which include the stick-breaking process with general stickbreaking weights, the classical Dirichlet process DP(a, H), the two-parameter Poisson-Dirichlet process PDP(a, b, H), the normalized inverse Gaussian process N-IG(a, H), the normalized generalized gamma process NGG(σ, a, H), and the generalized Dirichlet process GDP(a, r, H)
The method of moments used in this paper could be applied to the study of asymptotics for some Bayesian nonparametric posterior processes in the following situations: (i) when the parameter a is finite and the sample size is large; (ii) when the parameter a is large and the sample size is finite; (iii) when the parameter a and the sample size are both large

Summary

Introduction

Ever since the work of Ferguson (1973) the Dirichlet process has become a critical tool in Bayesian nonparametric statistics and has found applications in various areas, including machine learning, biological science, social science and so on. We are mainly concerned with three types of the asymptotics (strong law of large numbers, central limit theorem, and functional central limit theorem) for a number of processes, which include the stick-breaking process with general stickbreaking weights, the classical Dirichlet process DP(a, H) (see Ferguson, 1973), the two-parameter Poisson-Dirichlet process PDP(a, b, H) ( known as Pitman-Yor process, Pitman and Yor, 1997), the normalized inverse Gaussian process N-IG(a, H) (see Lijoi et al, 2005b), the normalized generalized gamma process NGG(σ, a, H) (see Lijoi et al, 2003, 2007; Brix, 1999), and the generalized Dirichlet process GDP(a, r, H) (see Lijoi et al, 2005a). Prior to this work the central limit theorem and the functional central limit theorem have been established only for the normalized inverse Gaussian process by using the finite dimensional distributions of the process itself. We follow the convention in the literature to continue to call them “processes”

Definitions

Moment results

Strong law of large numbers

Central limit theorems and functional central limit theorems

Numerical illustration

Concluding remarks