Anytime-Valid Evidence for Prespecified Predictive Corrections

Abstract

A predictive correction is a prespecified modification of an existing predictive distribution intended to reflect an anticipated change in future outcomes given their inputs, motivated, for example, by instrument recalibration, assay drift, or a known intervention. We study how to accumulate anytime-valid evidence that such a correction predicts incoming target outcomes better than the uncorrected source predictive distribution. A fixed nonnegative tilt transforms the source predictive into a corrected predictive, and the corrected-to-source predictive likelihood ratio is a conditional e-value whose running product forms an e-process. This process remains valid under optional stopping and arbitrary input sequences, including adaptively selected ones, while its logarithm equals the cumulative predictive log-score advantage of the correction. A conditional drift decomposition characterizes evidence growth under an arbitrary target predictive distribution, and a correction-dependent half-space identifies misspecified target distributions for which the same false-confirmation bound continues to hold. When the predictive likelihood ratio is strictly positive, its reciprocal yields an anytime-valid refutation boundary, while an overshoot identity explains why the realized null crossing probability may fall below the nominal level. Label-shift, conditional mean and variance, subgroup-specific, and exponential-family corrections arise as special cases. Prespecified mixtures accommodate uncertainty over corrections, predictable tilts permit adaptive betting, and beyond-tolerance comparisons target changes large enough to justify action. Cross-family calculations and synthetic experiments show that a boundary crossing supports the proposed correction relative to its reference but does not uniquely identify the mechanism responsible for the shift.