# Are there any results on the distribution of the posterior mean across data sets?

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)?

• This might not be exactly what you want, but the Bernstein von Mises thm (en.wikipedia.org/wiki/Bernstein%E2%80%93von_Mises_theorem) gives asymptotic normality of the posterior under certain regularity conditions. Jun 28 '18 at 18:39
• @marmle Thanks, yes I know. So is the distribution of means of normal posterior distributions across data sets normal? Probably... and does it hold in general? Jun 28 '18 at 18:41