  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.
