Threshold Martingales and the Evolution of Forecasts

Dean P. Foster; Robert A. Stine

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arXiv · 2105.06834

Threshold Martingales and the Evolution of Forecasts

Dean P. Foster·Robert A. Stine

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Abstract

This paper introduces a martingale that characterizes two properties of evolving forecast distributions. Ideal forecasts of a future event behave as martingales, sequen- tially updating the forecast to leverage the available information as the future event approaches. The threshold martingale introduced here measures the proportion of the forecast distribution lying below a threshold. In addition to being calibrated, a threshold martingale has quadratic variation that accumulates to a total determined by a quantile of the initial forecast distribution. Deviations from calibration or to- tal volatility signal problems in the underlying model. Calibration adjustments are well-known, and we augment these by introducing a martingale filter that improves volatility while guaranteeing smaller mean squared error. Thus, post-processing can rectify problems with calibration and volatility without revisiting the original forecast- ing model. We apply threshold martingales first to forecasts from simulated models and then to models that predict the winner in professional basketball games.

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Dean P. Foster, Robert A. Stine. 2021-05-14. Threshold Martingales and the Evolution of Forecasts. https://arxiv.org/abs/2105.06834

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