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 in the context of i.i.d. samples. 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. We also generalize 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 posterior consistency is proved in a model where the Kullback-Leibler condition cannot be satisfied by any prior. Other applications include Hellinger consistent density estimation in general mixture models with Dirichlet or Gibbs-type priors of full weak support.
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