We review de Finetti’s two coherence criteria for determinate probabilities: coherence1 defined in terms of previsions for a set of random variables that are undominated by the status quo – previsions immune to a sure-loss – and coherence2 defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence2 based on a generalization of Brier score for IP-forecasts that uses 1-sided, lower and upper, probability forecasts. However, whereas Brier score is a strictly proper scoring rule for eliciting determinate probabilities, we show that there is no real-valued strictly proper IP-score. Nonetheless, with respect to either of two decision rules – Gamma-Maximin or (Levi’s) E-admissibility + Gamma-Maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score.
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Teddy Seidenfeld
135J Baker Hall
Carnegie Mellon University
Pgh. PA 15213
Mark Schervish
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213-3890
USA
Joseph Kadane
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213
| Teddy Seidenfeld | teddy@stat.cmu.edu | |
| Mark Schervish | mark@stat.cmu.edu | |
| Joseph Kadane | kadane@stat.cmu.edu |
Send any remarks to isipta11@uibk.ac.at.