The asymptotic form of the Bayes acceptance region is derived for testing simple null-hypotheses in multiparameter exponential families. This result suggests a reasonable definition for tests which might be called "almost-Bayes". The rate at which the risk of the Bayes test converges to zero is obtained, showing the nature of its dependence on the prior distribution and providing a basis for comparison of almost-Bayes procedures. Concluding remarks contain a brief discussion of some asymptotic consequences of poor prior guessing.
"Asymptotic Behavior of Bayes Procedures for Testing Simple Hypotheses in Multiparameter Exponential Families." Ann. Statist. 6 (2) 346 - 361, March, 1978. https://doi.org/10.1214/aos/1176344129