In a previous paper, the authors introduced a new criterion of expectation consistency between probability distributions of discrete data given discrete parameter values and arbitrary posterior probability distributions for the parameter. It is here shown, under very weak assumptions, that expectation consistency implies that the posterior distributions are generalized Bayes. However when the posterior distributions are generalized Bayes, the implied prior distribution need not be unique. The class of implied distributions is characterized in terms of a partition of parameter space.
"Expectation Consistency and Generalized Bayes Inference." Ann. Statist. 1 (3) 478 - 485, May, 1973. https://doi.org/10.1214/aos/1176342413