On using "climatology" as a reference strategy in the Brier and ranked probability skill scores

The Brier and ranked probability skill scores are widely used as skill metrics of probabilistic forecasts of weather and climate. As skill scores, they compare the extent to which a forecast strategy outperforms a ( usually simpler) reference forecast strategy. The most widely used reference strategy is that of "climatology,'' in which the climatological probability ( or probabilities in the case of the ranked probability skill score) of the forecast variable is issued perpetually. The Brier and ranked probability skill scores are often considered harsh standards. It is shown that the scores are harsh because the expected value of these skill scores is less than 0 if nonclimatological forecast probabilities are issued. As a result, negative skill scores can often hide useful information content in the forecasts. An alternative formulation of the skill scores based on a reference strategy in which the outcome is independent of the forecast is equivalent to using randomly assigned probabilities but is not strictly proper. Nevertheless, positive values of the Brier skill score with random guessing as a strategy correspond to positive-sloping reliability curves, which is intuitively appealing because of the implication that the conditional probability of the forecast event increases as the forecast probability increases.