ESTIMATING A BINOMIAL PARAMETER: IS ROBUST BAYES REAL BAYES?

In robust Bayesian analysis, a prior is assumed to belong to a family instead of being specified exactly. The multiplicity of priors naturally leads to a collection of Bayes actions (estimates), and these often form a convex set (an interval in the case of a real parameter). It is clearly essential to be able to recommend one action from this set to the user. We address the following problem: if we systematically choose one action for each X thereby constructing a decision rule, is it going to be Bayes? Is it Bayes with respect to a prior in the original prior family? Even if it is not genuine Bayes, is it admissible?