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Abstract:
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Under model misspecification, it is known that usual Bayes posteriors often do not properly quantify uncertainty about the true or pseudo-true parameters. However, this lack of correct calibration is always defined with respect to a somewhat arbitrary notion of what is "true", making it conceptually unappealing and difficult to check empirically. We introduce a notion of internally coherent uncertainty quantification that is completely agnostic to the truth. Specifically, we consider the probability that two confidence sets constructed from independent data sets have nonempty overlap, and we establish a lower bound on this overlap probability that holds for any valid confidence sets. We show that, under misspecification, credible sets from the usual Bayes posterior can strongly violate this bound, indicating that it is not internally coherent.
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