The Dirichlet Process with Large Concentration Parameter

TL;DR

This paper investigates the asymptotic behavior of the Dirichlet process with large concentration parameter, showing convergence to a Brownian bridge and establishing related limit theorems relevant for Bayesian inference.

Contribution

It provides new theoretical results on the convergence of the Dirichlet process as the concentration parameter grows large, including a Glivenko-Cantelli theorem and quantile process convergence.

Findings

Convergence to a Brownian bridge as the concentration parameter increases.

A Glivenko-Cantelli type theorem for the Dirichlet process.

Weak convergence of the quantile process using the functional delta method.

Abstract

Ferguson's Dirichlet process plays an important role in nonparametric Bayesian inference. Let $P_a$ be the Dirichlet process in $\mathbb{R}$ with a base probability measure $H$ and a concentration parameter $a>0.$ In this paper, we show that $\sqrt {a} \big(P_a((-\infty,t]) -H((-\infty,t])\big)$ converges to a certain Brownian bridge as $a \to \infty.$ We also derive a certain Glivenko-Cantelli theorem for the Dirichlet process. Using the functional delta method, the weak convergence of the quantile process is also obtained. A large concentration parameter occurs when a statistician puts too much emphasize on his/her prior guess. This scenario also happens when the sample size is large and the posterior is used to make inference.