Truthfulness of Calibration Measures
This addresses a foundational issue in machine learning for forecasters and calibration systems, but it is incremental as it builds on existing calibration measures.
The paper tackles the problem of calibration measures in sequential prediction not being truthful, meaning forecasters can exploit them with poor forecasts, and introduces a new measure, Subsampled Smooth Calibration Error (SSCE), where truthful prediction is optimal up to a constant factor.
We initiate the study of the truthfulness of calibration measures in sequential prediction. A calibration measure is said to be truthful if the forecaster (approximately) minimizes the expected penalty by predicting the conditional expectation of the next outcome, given the prior distribution of outcomes. Truthfulness is an important property of calibration measures, ensuring that the forecaster is not incentivized to exploit the system with deliberate poor forecasts. This makes it an essential desideratum for calibration measures, alongside typical requirements, such as soundness and completeness. We conduct a taxonomy of existing calibration measures and their truthfulness. Perhaps surprisingly, we find that all of them are far from being truthful. That is, under existing calibration measures, there are simple distributions on which a polylogarithmic (or even zero) penalty is achievable, while truthful prediction leads to a polynomial penalty. Our main contribution is the introduction of a new calibration measure termed the Subsampled Smooth Calibration Error (SSCE) under which truthful prediction is optimal up to a constant multiplicative factor.