## Bernoulli

- Bernoulli
- Volume 24, Number 1 (2018), 231-256.

### Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures

Sophie Donnet, Vincent Rivoirard, Judith Rousseau, and Catia Scricciolo

#### Abstract

We provide conditions on the statistical model and the prior probability law to derive contraction rates of posterior distributions corresponding to data-dependent priors in an empirical Bayes approach for selecting prior hyper-parameter values. We aim at giving conditions in the same spirit as those in the seminal article of Ghosal and van der Vaart [*Ann. Statist.* **35** (2007) 192–223]. We then apply the result to specific statistical settings: density estimation using Dirichlet process mixtures of Gaussian densities with base measure depending on data-driven chosen hyper-parameter values and intensity function estimation of counting processes obeying the Aalen model. In the former setting, we also derive recovery rates for the related inverse problem of density deconvolution. In the latter, a simulation study for inhomogeneous Poisson processes illustrates the results.

#### Article information

**Source**

Bernoulli, Volume 24, Number 1 (2018), 231-256.

**Dates**

Received: March 2015

Revised: May 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142441

**Digital Object Identifier**

doi:10.3150/16-BEJ872

**Mathematical Reviews number (MathSciNet)**

MR3706755

**Zentralblatt MATH identifier**

06778326

**Keywords**

Aalen model counting processes Dirichlet process mixtures empirical Bayes posterior contraction rates

#### Citation

Donnet, Sophie; Rivoirard, Vincent; Rousseau, Judith; Scricciolo, Catia. Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures. Bernoulli 24 (2018), no. 1, 231--256. doi:10.3150/16-BEJ872. https://projecteuclid.org/euclid.bj/1501142441

#### Supplemental materials

- Supplement to “Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures”. This supplement contains the proofs of Theorem 2, Proposition 1 and Theorem 3 from the mentioned article.Digital Object Identifier: doi:10.3150/16-BEJ872SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.