Forecasting with Imprecise Probabilities

We review de Finetti's two coherence criteria for determinate probabilities: coherence(1) 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 coherence(2) defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence(2) 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.