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Elements of asymptotic theory with outer probability measures

Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as Bayesian inference, estimators or hypothesis testing need to be analysed in order to understand their behaviour, and motivate their use. In this article, we consider a class of outer measures based on the supremum of particular functions that we refer to as possibility functions. We then characterise the asymptotic behaviour of the corresponding Bayesian posterior uncertainties, from which the properties of the corresponding maximum a posteriori estimators can be deduced. These results are largely based on versions of both the law of large numbers and the central limit theorem that are adapted to possibility functions. Our motivation with outer measures is through the notion of uncertainty quantification, where verification of these procedures is of crucial importance. These introduced concepts shed a new light on some standard concepts such as the Fisher information and sufficient statistics and naturally strengthen the link between the frequentist and Bayesian approaches.