On the Bernstein-von Mises theorem for the Dirichlet process

TL;DR

This paper proves that the Laplace transforms of the posterior Dirichlet process converge to those of a Brownian bridge, providing a Bernstein-von Mises type result for Bayesian nonparametrics.

Contribution

It establishes uniform convergence of Laplace transforms of the posterior Dirichlet process to a Gaussian process, extending previous results to broader classes of functions.

Findings

Laplace transforms of the posterior Dirichlet process converge to those of a Brownian bridge.

Convergence holds uniformly over Glivenko-Cantelli classes.

Results are strengthened for real-valued variables and functions of bounded variation.

Abstract

We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.