Forecasting with imprecise probabilities

• Published in: International journal of approximate reasoning 2012-11, Vol.53 (8), p.1248-1261
• Main Authors: Seidenfeld, Teddy, Schervish, Mark J., Kadane, Joseph B.
• Format: Article
• Language: English
• Subjects: Applied sciences; Brier score; Coherence; Decision theory. Utility theory; Dominance; E-admissibility; Exact sciences and technology; Mathematics; Operational research and scientific management; Operational research. Management science; Probability and statistics; Probability theory and stochastic processes; Probability theory on algebraic and topological structures; Proper scoring rules; Sciences and techniques of general use; Γ-Maximin
• ISSN: 0888-613X; 1873-4731
• DOI: 10.1016/j.ijar.2012.06.018

We review de Finetti’s two coherence criteria for determinate probabilities: coherence1 defined in terms of previsions for a set of events 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 – Γ-maximin or (Levi’s) E-admissibility-+-Γ-maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score.