On the frequentist validity of Bayesian limits

Four frequentist theorems on the large-sample limit behaviour of posterior distributions are proved, for posterior consistency in metric or weak topologies; for posterior rates of convergence in metric topologies; for consistency of the Bayes factor for hypothesis testing or model selection; and a new theorem that explains how credible sets are to be transformed to become asymptotic confidence sets. Proofs require the existence of suitable test sequences and priors that give rise to a property of local prior predictive distributions which generalizes Schwartz's Kullback-Leibler condition as a weakened form of Le~Cam's contiguity. Results are applied in a range of examples and counterexamples.