Optimally robust credible sets with a class of priors

We determine a credible set A that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ of the starting prior π0(θ). Among the class of sets with credibility γ under π0, the “optimally robust” set will be the one which maximizes the minimum probability of including θ as the prior varies over Γ. This procedure is also Γ-minimax with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ, the ε-contamination class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.