Maximum a posteriori estimator is a Bayes estimator under the 0-1 loss function and some given prior distribution. I was wondering how to give an estimation that is best in some sense if the prior distribution is also random?
For example, $T_1(X)$ is the maximum a posteriori estimator of parameter $\theta$ under some prior distribution $p_1$ on $\theta$, $T_2(X)$ is the maximum a posteriori estimator of parameter $\theta$ under some prior distribution $p_2$ on $\theta$, and the prior distribution $p_1$ occurs with probability $q$ and $p_2$ with $1-q$.
How shall one determine the best non-randomized estimator in some kind of sense?
How shall one determine the best randomized estimator in some kind of sense? A "randomized" estimator is represented by a distribution on the set of all possible estimators.
Are there some references? Thanks and regards!