If there are many independent, identically distributed observations governed by a smooth, finite-dimensional statistical model, the Bayes estimate and the maximum likelihood estimate will be close. Furthermore, the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth. Thus, Bayesian confidence sets have good frequentist coverage properties, and conversely. However, even for the simplest infinite-dimensional models, such results do not hold. The object here is to give some examples.
"Wald Lecture: On the Bernstein-von Mises theorem with infinite-dimensional parameters." Ann. Statist. 27 (4) 1119 - 1141, August 1999. https://doi.org/10.1214/aos/1017938917