Summary
Exploiting the notion of identifiability of mixtures of exponential families with respect to a vector parameter θ, it is shown that the posterior expectation of θ characterizes the prior distribution of θ. The result is applied to normal and negative multinomial distributions.
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Cacoullos, T. Characterizing priors by posterior expectations in multiparameter exponential families. Ann Inst Stat Math 39, 399–405 (1987). https://doi.org/10.1007/BF02491477
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DOI: https://doi.org/10.1007/BF02491477