Asymmetric penalties underlie proper loss functions in probabilistic forecasting

原文摘要

Abstract Accurately forecasting the probability distribution of phenomena of interest is a classic and increasingly widespread goal in statistics and decision theory. In comparison to point forecasts, probabilistic forecasts aim to provide a more complete and informative characterization of the target variable. This endeavour is only fruitful, however, if a forecast is ‘close’ to the distribution it attempts to predict. The role of a loss function, also known as a scoring rule, is to make this precise by providing a quantitative measure of proximity between a forecast distribution and the target random variable. Numerous loss functions have been proposed in the literature, with a strong focus on proper losses, that is, losses whose expectations are minimized when the forecast distribution is the same as the target. In this paper, we show that a broad class of proper loss functions penalize asymmetrically, in the sense that underestimating a given parameter of the target distribution can incur a larger loss than overestimating it, or vice versa. Our theory covers many popular losses, such as the logarithmic, continuous ranked probability, quadratic and spherical losses, as well as the energy and threshold-weighted generalizations of continuous ranked probability loss. To complement our theory, we present experiments with real epidemiological, meteorological and retail forecast datasets. Furthermore, as an implication of the loss asymmetries revealed by our work, we show that hedging is possible under distribution shift.

摘要来源：Crossref