This paper develops inference and statistical decision for set-identified parameters from the robust Bayes perspective. When a model is set-identified, prior knowledge for model parameters is decomposed into two parts: the one that can be updated by data (revisable prior knowledge) and the one that never be updated (unrevisable prior knowledge.) We introduce a class of prior distributions that shares a single prior distribution for the revisable, but allows for arbitrary prior distributions for the unrevisable. A posterior inference procedure proposed in this paper operates on the resulting class of posteriors by focusing on the posterior lower and upper probabilities. We analyze point estimation of the set-identified parameters with applying the gamma-minimax criterion. We propose a robustified posterior credible region for the set-identified parameters by focusing on a contour set of the posterior lower probability. Our framework offers a procedure to eliminate set-identified nuisance parameters, and yields inference for the marginalized identified set. For an interval identified parameter case, we establish asymptotic equivalence of the lower probability inference to frequentist inference for the identified set.

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