Angelo Gilio, Giuseppe Sanfilippo

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Coherent conditional probabilities and proper scoring rules

Abstract

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.

Keywords

Conditional probability assessments, coherence, penalty criterion, proper scoring rules, conditional scoring rules, weak dominance, strong dominance, admissibility, Bregman divergence, g-coherence, total coherence, imprecise probability assessments.

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Authors’ addresses

Angelo Gilio
Dipartimento di Metodi e Modelli Matematici
Via A. Scarpa 16, 00161 Roma (Italy)

Giuseppe Sanfilippo
Dipartimento di Scienze Statistiche e Matematiche ''S.Vianelli''
Universita' degli Studi di Palermo
Viale delle Scienze ed.13
90128 Palermo,Italy

E-mail addresses

Angelo Gilio		gilio@dmmm.uniroma1.it
Giuseppe Sanfilippo		sanfilippo@unipa.it