Scoring Rules for Belief Functions and Imprecise Probabilities: A Comparison

This paper investigates de Finetti's coherence as an operational 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 probability are undistinguishable, which is surprising given that Lower probabilities are known to exist which do no satisfy the axioms of Belief functions. The main contribution of this paper consists in putting forward 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 comparison we show that scoring rules allow us to distinguish coherence-wise between Belief functions and Imprecise probabilities.