Conditions are given under which a sequence of posterior distributions converges weakly to a degenerate distribution. It is not assumed that the model being used actually governs the data. Exponential models are studied in particular and are shown to be well behaved under mild assumptions. Large-deviation results for $U$-statistics and posterior odds are also given.
Robert H. Berk. "Consistency a Posteriori." Ann. Math. Statist. 41 (3) 894 - 906, June, 1970. https://doi.org/10.1214/aoms/1177696967