In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.
"Asymptotic Behavior of Bayes Tests and Bayes Risk." Ann. Statist. 2 (2) 278 - 294, March, 1974. https://doi.org/10.1214/aos/1176342663