Proper scoring rules for interval probabilistic forecasts

Interval probabilistic forecasts for a binary event are forecasts issued as a range of probabilities for the occurrence of the event, for example, ‘chance of rain: 10-20%’. To verify interval probabilistic forecasts, use can be made of a scoring rule that assigns a score to each forecast-outcome pair. An important requirement for scoring rules, if they are to provide a faithful assessment of a forecaster, is that they be proper, by which is meant that they direct forecasters to issue their true beliefs as their forecasts. Proper scoring rules for probabilistic forecasts issued as precise numbers have been studied extensively. But, applying such a proper scoring rule to, for example, the mid-point of an interval probabilistic forecast, does not, typically, produce a proper scoring rule for interval probabilistic forecasts. Complementing parallel work by other authors, we derive a general characterisation of scoring rules that are proper for interval probabilistic forecasts and from this characterisation we determine particular scoring rules for interval probabilistic forecasts that correspond to the familiar scoring rules used for probabilistic forecasts given as precise probabilities. All the scoring rules we derive apply immediately to rounded probabilistic forecasts, being a special case of interval probabilistic forecasts.