Exact grid-free confidence-region computation from dependent p-value functions under arbitrary dependence

Abstract

Repeated sample splitting, cross-fitting, conformal ensembling, and related randomized workflows often produce multiple dependent valid p-value functions for the same target. Existing p-merging theory guarantees pointwise validity under arbitrary dependence, but turning the merged output into a confidence region typically requires repeated evaluation on a grid over the parameter space. This inversion step can be computationally costly, approximation-dependent, and increasingly difficult to scale. We study when confidence regions can instead be computed exactly from split-wise regions without gridding the parameter space. Our main structural result shows that mergers induced by step calibrators form a broad exactly executable class at the region level, and a partial converse within the calibrator-induced family indicates that exact executability is closely tied to step structure. Within this class, we develop exact voting algorithms, including an adaptive multi-quantile contour aggregator that avoids pre-specifying a single order-statistic threshold while preserving finite-sample validity under arbitrary dependence. Simulation studies on repeated-split regression and conformal prediction, together with real-data regression examples, show that the proposed methods provide stable, robustness-oriented exact inference with substantial runtime gains over grid-inversion comparators. A grid-resolution benchmark shows that fixed-k voting and adaptive multi-quantile voting are essentially insensitive to inversion-grid refinement, while a multidimensional single-threshold stress test illustrates the dimensional blow-up faced by grid-inversion baselines in a simple box-geometry setting. Taken together, the results show that arbitrary-dependence contour merging can be turned into an exactly executable region-computation framework rather than merely a pointwise validity device.