This paper reconciles the asymptotic disagreement between Bayesian and frequentist inference in set-identified models by adopting a multiple-prior (robust) Bayesian approach. We propose new tools for Bayesian inference in set-identified models and show that they have a well-defined posterior interpretation in finite samples and are asymptotically valid from the frequentist perspective. The main idea is to construct a prior class that removes the source of the disagreement: the need to specify an unrevisable prior for the structural parameter given the reduced-form parameter. The corresponding class of posteriors can be summarized by reporting the ‘posterior lower and upper probabilities’ of a given event and/or the ‘set of posterior means’ and the associated ‘robust credible region’. We show that the set of posterior means is a consistent estimator of the true identified set and the robust credible region has the correct frequentist asymptotic coverage for the true identified set if it is convex. Otherwise, the method provides posterior inference about the convex hull of the identified set. For impulse-response analysis in set-identified Structural Vector Autoregressions, the new tools can be used to overcome or quantify the sensitivity of standard Bayesian inference to the choice of an unrevisable prior.

中文翻译：

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集识别模型的鲁棒贝叶斯推理

本文通过采用多先验（稳健）贝叶斯方法，调和了集合识别模型中贝叶斯推理和频率推理之间的渐近分歧。我们为集合识别模型中的贝叶斯推理提出了新工具，并表明它们在有限样本中具有明确定义的后验解释，并且从频率论的角度来看是渐近有效的。主要思想是构建一个先验类来消除分歧的根源：需要为给定简化形式参数的结构参数指定一个不可修改的先验。可以通过报告给定事件的“后验下限和上限概率”和/或“后验平均值集”和相关联的“可靠可信区域”来总结相应的后验类别。我们表明后验均值集是真实识别集的一致估计量，并且鲁棒可信区域对真实识别集具有正确的频率渐近覆盖，如果它是凸的。否则，该方法提供关于已识别集合的凸包的后验推断。对于集合识别结构向量自回归中的脉冲响应分析，新工具可用于克服或量化标准贝叶斯推理对不可修正先验选择的敏感性。