The formalism of operational statistics, a generalized approach to probability and statistics, provides a setting within which inference strategies can be studied with great clarity. This paper is concerned with the asymptotic behavior of the Bayesian inference strategy in this setting. We consider a sequence of posterior distributions, obtained from a prior as a result of successive conditionings by the events of an admissible sequence. We identify certain statistical hypotheses whose limiting posterior probabilities converge to one. We describe these hypotheses, (...) and show that when the prior is vague, they contain those probability models which represent the long-run relative frequencies of occurrence for the events in the sequence. (shrink)