LAUSR

When probability forecasts are made of a binary event, a commonly used measure for assessing the forecasts is the Brier score. One of its properties is that it is proper, meaning that its expected value cannot be improved by the forecaster issuing a probability other than his/her true belief. This property assumes that the occurrence or otherwise of the forecast event is recorded without error. This note investigates what forecast should be made in order to minimize the expected value of the Brier score when errors are present in the observations. Should it still be the forecaster's true belief or should it be something else, implying that the forecaster should hedge his/her forecast? The answer is that it depends on whether the forecaster can model the error mechanism or whether the error mechanism is unknown. It is shown that in the former case the forecaster's true belief of the probability of the event should still be forecast. However, in the case of an unknown error mechanism, the forecaster should attempt to forecast the probability that the erroneous observation indicates that the event has occurred, rather than the true probability of the event.