Nobel laureate Niels Bohr is famously quoted as saying, “Prediction is very difficult, especially if it’s about the future.” The science (or perhaps the art) of forecasting is no easy task and lends itself to a large amount of uncertainty. For this reason, practitioners interested in prediction have increasingly migrated to probabilistic forecasting, where an entire distribution is given as the forecast instead of a single number, thus fully quantifying the inherent uncertainty. In such a setting, traditional metrics of assessing and comparing predictive performance, such as mean squared error (MSE), are no longer appropriate. Instead, proper scoring rules are utilized to evaluate and rank forecast methods. A scoring rule is a function that takes a predictive distribution along with an observed value and outputs a real number called the score. Such a rule is said to be proper if the expected score is maximized when the predictive distribution is the same as the distribution from which the observation was drawn.
For quality assessments in reliability and industrial engineering, it is often necessary to predict the number of future events (e.g., system or component failures). Examples include the prediction of warranty returns and the prediction of future product failures that could lead to serious property damages and/or human casualties. Business decisions such as a product recall are based on such predictions.