  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.
