Uncertainty about the choice of identifying assumptions is common in causal studies, but has been often ignored in empirical practice. This paper considers uncertainty over a class of models that impose different sets of identifying assumptions, which, in general, leads to a mix of point- and set-identified models. We propose a method for performing inference in the presence of this type of uncertainty by generalizing Bayesian model averaging. Our proposal is to consider ambiguous belief (multiple posteriors) for the set-identified models, and to combine them with a single posterior in a model that is either point-identified or that imposes non-dogmatic identifying assumptions in the form of a Bayesian prior. The output is a set of posteriors ( post-averaging ambiguous belief ) that are mixtures of the single posterior and any element of the class of multiple posteriors, with mixture weights the posterior probabilities of the models. We propose to summarize the post-averaging ambiguous belief by reporting the range of posterior means and the associated credible regions, and offer a simple algorithm to compute these quantities. We establish conditions under which the data are informative about model probabilities, which occurs when the models are distinguishable for some distribution of data and/or specify different priors for reduced-form parameters, and examine the asymptotic behavior of the posterior model probabilities. The method is general and allows for dogmatic and non-dogmatic identifying assumptions, multiple point-identified models, multiple set-identified models, and nested or non-nested models.

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