Convergence Rates of Nonparametric Posterior Distributions

TL;DR

This paper develops general theorems for the convergence rates of nonparametric posterior distributions, extending previous results and introducing new tools like Hausdorff α-entropy and an improved prior concentration concept.

Contribution

It provides a unified framework for analyzing posterior convergence rates in nonparametric models, extending prior theoretical results with new mathematical tools.

Findings

Established general posterior convergence rate theorems.

Extended previous results by Ghosal, Van der Vaart, Shen, Wasserman, Walker, Lijor, and Prunster.

Applied the theorems to various statistical models.

Abstract

We study the asymptotic behavior of posterior distributions. We present general posterior convergence rate theorems, which extend several results on posterior convergence rates provided by Ghosal and Van der Vaart (2000), Shen and Wasserman (2001) and Walker, Lijor and Prunster (2007). Our main tools are the Hausdorff $\alpha$-entropy introduced by Xing and Ranneby (2008) and a new notion of prior concentration, which is a slight improvement of the usual prior concentration provided by Ghosal and Van der Vaart (2000). We apply our results to several statistical models.