I know that this question borderlines on Bayesian and frequentist philosophy, somewhat related to this question.
Bayesian point estimation sometimes uses the mean of the posterior distribution. That is, the mean of the distribution of a parameter conditional on the data. True Bayesians would update the posterior when new data become available. However, I am interested in the alternative situation when a new data set (let's say of equal size as a first data set) is used to compute a second posterior mean. If we repeated this process with many new samples of equal size, we get a distribution of posterior means.
Are there results on the form of this distribution? In particular, is this distribution normal (can we apply something like a central limit theorem on the posterior means)?