Coherent conditional probabilities and proper scoring rules

In this paper we study the relationship between the notion of coherence for conditional probability assessments on a family of conditional events and the notion of admissibility with respect to scoring rules. By extending a recent result given in literature for unconditional events, we prove, for any given strictly proper scoring rule s, the equivalence between the coherence of a conditional probability assessment and its admissibility with respect to s. In this paper we focus our analysis on the case of continuous bounded scoring rules. In this context a key role is also played by Bregman divergence and by a related theoretical aspect. Finally, we briefly illustrate a possible way of defining (generalized) coherence of interval-valued probability assessments by exploiting the notion of admissibility given for precise probability assessments.