On the Bernstein-von Mises theorem with infinite-dimensional parameters

If there are many independent, identically distributed observations governe d 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 clos e to the distribution of the maximum likelihood estimate around truth. Thus , Bayesian confidence sets have good frequentist coverage properties, and c onversely. However, even for the simplest infinite-dimensional models, such results do not hold. The object here is to give some examples.