Brier Loss Function

Let ⚠ $\Omega$ be a finite set and ⚠ $\Gamma$ be the set of all probability measures on ⚠ $\Omega$. For the definition and properties of the Brier loss function (first defined by Brier in 1950), see Vovk and Zhdanov (2008). The latter paper Prediction with expert advice for the Brier game gives an optimal algorithm for prediction with expert advice with this loss function.

References

• Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78, 1–3.
• Vovk, Vladimir and Zhdanov, Fedor. Prediction with expert advice for the Brier game. Proceedings of ICML 2008 and arXiv technical report.