On Bayesian consistency Stephen Walker University of Bath, UK and Nils Lid Hjort University of Oslo, Norway [Received April 2000. Revised May 2001] Summary. We consider a sequence of posterior distributions based on a data-dependent prior which we shall refer to as a pseudoposterior distribution) and establish simple conditions under which the sequence is Hellinger consistent. It is shown how investigations into these pseudo- posteriors assist with the understanding of some true posterior distributions, including Po Âlya trees, the in®nite dimensional exponential family and mixture models. Keywords: Asymptotics; Bayesian sieve; Bayes nonparametrics; Consistency 1. Introduction Asymptotics play an important role in statistics. In classical density estimation this role is crucial, providing justi®cations for a wide range of nonparametric estimators such as kernel- based estimators and sieve maximum likelihood estimators Shen and Wong, 1994; Wong and Shen, 1995) and other nonparametric estimators see, for example, van de Geer 1993)). Establishing consistency and rates of convergence with respect to a suitable metric, typically the Hellinger distance, is the key point to this area of research. However, Bayesian nonparametric methods have only recently started to undergo asymp- totic studies. Early work was done by Schwartz 1965) who established that a prior which puts positive mass on all Kullback±Leibler neighbourhoods of the true density is weakly consistent. Previously, Freedman 1963), and later Diaconis and Freedman 1986) in a more natural setting, had demonstrated that priors which put positive mass on all weak neigh- bourhoods of the true distribution function are not necessarily weakly consistent. Recent attention has switched to studying and ®nding suf®cient conditions for strong Hellinger) consistency. Suppose that  is a prior distribution on the set of all probability densities over an interval or region of interest. As data x 1 , x 2 , ... accumulate from some unknown underlying density f 0 , will the Bayesian posterior distribution  n  .   . jx 1 ,..., x n  concentrate around this f 0 ? Barron et al. 1999) presented one such Bayesian nonparametric consistency theorem; the corresponding theorem of Ghosal et al. 1999a) is of a similar nature. Barron et al. 1999) made two assumptions to prove consistency in the Hellinger metric, i.e.  n f f : H f  >g! 0 almost surely Address for correspondence: Stephen Walker, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. E-mail:massgw@maths.bath.ac.uk & 2001 Royal Statistical Society 1369±7412/01/63811 J. R. Statist. Soc. B 2001) 63, Part 4, pp.811±821