For statistical problems based on groups, it is shown that there is convergence in probability to an invariant posterior distribution if and only if the posterior corresponds to right-invariant Haar prior and the group can support an asymptotically right-invariant sequence of proper prior distributions.
M. Stone. "Necessary and Sufficient Condition for Convergence in Probability to Invariant Posterior Distributions." Ann. Math. Statist. 41 (4) 1349 - 1353, August, 1970. https://doi.org/10.1214/aoms/1177696911