Criteria for posterior consistency and convergence at a rate

Abstract

Frequentist conditions for asymptotic consistency of Bayesian procedures with i.i.d. data 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 these criteria. We derive a versatile new posterior consistency theorem, which is used to consider Kullback-Leibler consistency and indicate when it is sufficient to have a prior that charges metric balls instead of KL-neighbourhoods. We generalize our proposal to sieved models with Barron’s negligible prior mass condition and to separable models with variations on Walker’s condition. Results are also applied in semi-parametric consistency: support boundary estimation is considered explicitly and consistency is proved in a model for which Kullback-Leibler priors do not exist. As a further demonstration of applicability, we consider metric consistency at a rate: under a mild integrability condition, the second-order Ghosal-Ghosh-van der Vaart prior mass condition can be relaxed to a lower bound for ordinary KL-neighbourhoods. The posterior rate is derived in a parametric model for heavy-tailed distributions in which the Ghosal-Ghosh-van der Vaart condition cannot be satisfied by any prior.