Truthfulness of Calibration Measures

TL;DR

This paper studies the truthfulness of calibration measures in sequential prediction, revealing that existing measures are not truthful and introducing a new measure, SSCE, that encourages truthful forecasting.

Contribution

It provides a taxonomy of calibration measures, demonstrates their lack of truthfulness, and proposes the SSCE as a truthful calibration measure.

Findings

Existing calibration measures are not truthful.

Simple distributions can exploit current measures for low penalties.

SSCE makes truthful prediction nearly optimal.

Abstract

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.