Asymmetric Penalties Underlie Proper Loss Functions in Probabilistic Forecasting

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ID: 319160
2026
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Abstract
Summary Accurately forecasting the probability distribution of phenomena of interest is a classic and ever more 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 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 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 data sets. Further, as an implication of the loss asymmetries revealed by our work, we show that hedging is possible under a setting of distribution shift.
Reference Key
openalex_W4414769611 Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Erez Buchweitz, João Vitor Romano, Ryan J. Tibshirani
Journal jurnal biometrika dan kependudukan
Year 2026
DOI
10.1093/biomet/asag042
URL
https://doi.org/10.1093/biomet/asag042
Keywords Keywords not found

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