Abstract

It is now widely agreed that one should assess the performance of Bayesian estimators of infinite dimensional parameters by the asymptotic behaviour of the posterior distribution. A rich literature has developed over the last 10 years, aimed at establishing weak and easily verifiable conditions on the prior such that the posterior concentrates almost surely on suitably defined neighbourhoods of the true parameter. Establishing tight bounds on the size of these shrinking neighbourhoods under various nonparametric priors has also been a concern. Omitting technical details, I will briefly outline the density estimation problem and the usual sieve constructions that, together with a well known support condition, provide sufficient conditions for strong consistency of the posterior. I will then present alternative approaches due to Walker (2004), which do not rely on the challenging sieve constructions of previous work, but rather on a simple and elegant martingale argument. I will set the paper in the context of subsequent work, highlighting – among others – the paper of Walker, Lijoi and Prunster (2007), which develops one of Walker’s arguments in order to improve the previously known convergence rates under two popular priors.

Links to relevent papers:

Walker. Ann. Statist. 32 (2004), no. 5, 2028—2043.

Walker et. al. Ann. Statist. 35 (2007), no. 2, 738—746.

Ghosal and Tang. J. Statist. Plann. Inference 137 (2007), no. 6, 1711—1726.