Quantile Merging for Bootstrapped Transductive Conformal Prediction

Transductive conformal prediction constructs joint prediction sets for groups of test samples, but practical implementations face a fundamental challenge: calibration data are typically available as i.i.d. individual samples, whereas inference is performed on groups whose size may vary at deployment time. We study how to construct transductive calibration sets from a fixed pool of calibration samples while preserving coverage. We consider bootstrap sampling from the empirical calibration distribution and show that valid marginal coverage requires an adjustment of the calibration quantile. To address the variance increase from bootstrap sampling, we introduce a $K$-fold transductive conformal framework that repeats bootstrap calibration and aggregates the resulting predictors using p-value and e-value merging techniques. We show that all considered aggregation schemes admit an equivalent quantile-level implementation, reducing inference to standard split conformal prediction with a merged calibration quantile. We further theoretically characterize the effects of calibration size, group size, and the number of folds on coverage and efficiency. Empirical results validate the theory and demonstrate substantial improvements in stability over single fold bootstrap calibration.