Criteria for posterior consistency

Abstract

Frequentist conditions for asymptotic suitability of Bayesian procedures focus on lower bounds for prior mass in Kullback-Leibler neighbourhoods of the data distribution. The goal of this paper is to investigate the flexibility in criteria for posterior consistency with i.i.d. data. We formulate a versatile posterior consistency theorem that applies both to well- and mis-specified models and which we use to re-derive Schwartz's theorem, consider Kullback-Leibler consistency and formulate consistency theorems in which priors charge metric balls. It is generalized to sieved models with Barron's negligible prior mass condition and to separable models with variations on Walker's consistency theorem. Results also apply to marginal semi-parametric consistency: support boundary estimation is considered explicitly and consistency is proved in a model for which Kullback-Leibler priors do not exist. Other examples include consistent density estimation in mixture models with Dirichlet or Gibbs-type priors of full weak support. Regarding posterior convergence at a rate, it is shown that under a mild integrability condition, the second-order Ghosal-Ghosh-van der Vaart prior mass condition can be relaxed to a lower bound to the prior mass in Schwartz's Kullback-Leibler neighbourhoods. The posterior rate of convergence is derived in a simple, parametric model for heavy-tailed distributions in which the Ghosal-Ghosh-van der Vaart condition cannot be satisfied by any prior.