Forecasting with Imprecise Probabilities

<p>We review de Finetti’s two coherence criteria for determinate probabilities: <em>coherence</em><sub>1</sub>defined in terms of previsions for a set of events that are undominated by the status quo – previsions immune to a sure-loss – and <em>coherence</em><sub>2</sub> defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence<sub>2</sub> 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 <em>real-valued</em>strictly proper IP-score. Nonetheless, with respect to either of two decision rules – Γ-<em>maximin</em> or (Levi’s) E-a<em>dmissibility</em>-+-Γ-<em>maximin</em> – we give a <em>lexicographic</em> strictly proper IP-scoring rule that is based on Brier score.</p>