This paper extends some of the results obtained by Freedman . In Section 1 a class of prior distributions on the space of all substochastic distributions on the positive integers is given, such that along almost all sample sequences the corresponding posterior distributions of the expectations of all bounded functions on the positive integers are asymptotically normal. Section 2 shows that most of Freedman's results carry over to the case of distributions on the closed unit interval.
"Asymptotic Behavior of Bayes' Estimates." Ann. Math. Statist. 35 (2) 846 - 856, June, 1964. https://doi.org/10.1214/aoms/1177703584