I am wondering about the different ways that Bayesian and Frequentist statistic connect with each other.
I recalled that the Maximum Likelihood estimate of a parameter $\theta$ is not necessarily an unbiased estimator of that parameter.
That made me wonder: Is the Mean Posterior estimate of $\theta$ an unbiased estimator?
Does $\phi(x)=E(\theta|x)$, imply $E(\phi(x)|\theta)=\theta$?
Note that this is indeed a meaningful question, since $\phi(x)$, while it is a Bayesian estimator, is simply a function from the data to the real line and so can also be seen as a classical frequentist estimator.
If this question cannot be answered in general, please assume the prior is uniform.
If not, is there some other Bayesian estimator (i.e. a function from the posterior to $\mathbb R$) that is always an unbiased estimator in the frequentist sense?