Iterated Bayesian Updates

Question

I get a sequence of data that is generated by a distribution with parameter $a_0$ (e.g. $\mathcal{N}(a_0,1)$). I assume a prior distribution $P(a)$, and Bayesian update for the belief according to the data. Is it true that my belief distribution will converge to a single point distribution on $a_0$?

Answer 1

It depends on the original distribution, and the strength of the prior.

For example, if the original distribution is $\mathcal{N}(0,1)$, and the prior is $\mathcal{N}(100,.1)$, then almost surely you will never converge to the true distribution.