Parameter estimation with a class of outer probability measures

We explore the interplay between random and deterministic phenomena using a representation of uncertainty based on the measure-theoretic concept of outer measure. The meaning of the analogues of different probabilistic concepts is investigated and examples of application are given. The novelty of this article lies mainly in the suitability of the tools introduced for jointly representing random and deterministic uncertainty. These tools are shown to yield intuitive results in simple situations and to generalise easily to more complex cases. Connections with Dempster-Shafer theory, the empirical Bayes methods and generalised Bayesian inference are also highlighted.