On the frequentist validity of Bayesian limits

To the frequentist that computes posteriors, not all priors are useful asymptotically: in this paper Schwartz's Kullback-Leibler condition is generalized to prove four simple and fully general frequentist theorems on the large-sample limit behaviour of posterior distributions, for posterior consistency; for posterior rates of convergence; for consistency of the Bayes factor in hypothesis testing or model selection; and a new theorem that explains how credible sets are converted to asymptotic confidence sets. Proofs require the existence of a Bayesian type of test sequence and priors that give rise to local prior predictive distributions that satisfy a weakened form of Le~Cam's contiguity with respect to the data distribution. Results are applied in a wide range of examples and counterexamples.