Scoring Rules for Belief Functions and Imprecise Probabilities: A Comparison

This paper investigates de Finetti’s coherence as an opera- tional foundation for a wide range of non-additive uncertainty measures and focuses, in particular, on Belief functions and Lower probabilities. In a companion paper we identify a number of non-limiting circumstances under which Dutch Book criteria for Belief functions and Lower proba- bility are undistinguishable, which is surprising given that Lower prob- abilities are known to exist which do no satisfy the axioms of Belief functions. The main contribution of this paper consists in putting for- ward a comparison between a criterion based on the Brier scoring rule for Belief Functions and the scoring rule introduced in 2012 by Seidenfeld, Schervish and Kadane for Imprecise probabilities. Through this compar- ison we show that scoring rules allow us to distinguish coherence-wise between Belief functions and Imprecise probabilities.