Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures

TL;DR

This paper establishes general conditions for posterior concentration rates in empirical Bayes methods and applies them to density estimation with Dirichlet process mixtures and intensity estimation for counting processes, supported by simulations.

Contribution

It provides near-tractable conditions for posterior concentration in empirical Bayes settings and applies these to new models including Dirichlet process mixtures and Aalen counting processes.

Findings

Derived posterior concentration rates for Dirichlet process mixtures.

Established conditions for counting process intensity estimation.

Validated results through simulation studies.

Abstract

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.