On the scoring approach to admissibility of uncertainty measures in expert systems

Abstract

This paper arose from our need to rigorize, clarify, and address fully the results of Lindley's paper (Scoring rules and the inevitability of probability, Internat. Statist. Rev. 50, (1982), 1–26). Herein, we develop a calculus of admissibility in a game theoretic setting. In the case of an additive aggregation function, it is shown that decomposable measures, such as those used in fuzzy logics, are admissible. Also, the problem of the admissibility of the Dempster-Shafer belief functions is investigated via the concept of random sets. It is shown that the class of admissible measures in a scoring framework depends on the assumptions concerning the aggregation function in use. In particular, for nonadditive aggregation functions, an admissible measure may not be transformable to a finitely additive probability measure.