Under data shift, prediction intervals can be provably shortest on average

Zhongyang Li proves that inverse conditional-density scores with randomized weighted calibration minimize expected interval length for every calibration sample size, and that estimated versions stay near-optimal after selection.

Academic

Zhongyang Li

University of Connecticut

Research Digest·30 October 2026·2 min read

Illustration: Under data shift, prediction intervals can be provably shortest on average

In one line

Among all valid rules for a fixed calibration sample size, randomized weighted conformal calibration with inverse conditional-density scores minimizes expected prediction-set length under known covariate shift.

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An idea this paper builds on

conformal predictionA method that produces prediction sets with guaranteed coverage, enabling risk control without distributional assumptions.5 papers →

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Deploying a model where typical inputs differ from training, you want short prediction ranges that usually contain the true answer. Standard , which tunes ranges from calibration data, assumes old and new inputs are similar.

This work assumes the input mix change is known as a probability ratio. For any fixed calibration size, the shortest valid ranges come from scoring each possible response by inverse conditional density, then randomized weighted calibration. The ratio appears only in calibration.

Main finding: this rule minimizes expected range length on the new distribution among all valid rules, even with small samples. Density estimation errors grow linearly, and selecting estimators by resampled length preserves the guarantee. This enables shorter trustworthy intervals under known shift.

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Concepts

conformal prediction weighted conformal calibration covariate shift inverse conditional density score conformal predictor selection kernel histogram density estimation