Truthful Calibration Measures for Sequential Prediction

arXiv:2608.21348v2 Announce Type: replace-cross Abstract: Calibration requires probabilistic reports to be conditionally unbiased and reliably interpretable as probabilities. A calibration measure assigns numerical error to miscalibrated reports. Haghtalab et al. (2024) proposed an approximately truthful calibration measure for online prediction, leaving open whether exact truthfulness is compatible with completeness and soundness. We resolve this question negatively for sequential binary prediction: exact truthfulness is incompatible with completeness and soundness, even for independent outcomes. We then show that this impossibility is specific to exact truthfulness. On the positive side, we give two general reductions that transform a base calibration measure satisfying a uniform completeness condition into additively and multiplicatively approximately truthful measures, respectively. Applying the multiplicative reduction, for every $0<\varepsilon<1$ we construct a sound and complete calibration measure that is $1+\exp\left(-\Omega(T^{1-\varepsilon})\right)$-multiplicatively truthful. This improves the approximate truthfulness guarantee of Haghtalab et al. (2024). We also prove a general transfer theorem showing that the uniform completeness of one calibration measure transfers to another when they are quantitatively related. Using this result, we show that every consistent calibration measure (Blasiok et al., 2023) satisfies uniform completeness and can serve as the base calibration measure to achieve $1 + o(1)$ multiplicative truthfulness.