Multi-Scale Conformal Prediction: A Theoretical Framework with Coverage Guarantees
Ali Baheri, Marzieh Amiri Shahbazi
TL;DR
This work extends conformal prediction to a multi-scale setting by defining scale-specific conformity scores and constructing a final prediction set as the intersection of scale-wise sets. By distributing the allowed miscoverage $\alpha$ across scales and analyzing both independence and dependence across scales, the approach retains finite-sample marginal coverage while often yielding smaller, more informative prediction sets. Theoretical results establish coverage validity, efficiency gains from intersection, an optimal allocation principle based on scale informativeness, and asymptotic optimality to a minimal conditional-coverage set. Numerical experiments on synthetic, multi-scale data demonstrate that multi-scale conformal prediction maintains nominal coverage and improves efficiency relative to single-scale methods. The framework opens avenues for applying conformal inference to hierarchical and multi-resolution data domains, with potential extensions to adaptive scale definitions and allocation strategies.
Abstract
We propose a multi-scale extension of conformal prediction, an approach that constructs prediction sets with finite-sample coverage guarantees under minimal statistical assumptions. Classic conformal prediction relies on a single notion of conformity, overlooking the multi-level structures that arise in applications such as image analysis, hierarchical data exploration, and multi-resolution time series modeling. In contrast, the proposed framework defines a distinct conformity function at each relevant scale or resolution, producing multiple conformal predictors whose prediction sets are then intersected to form the final multi-scale output. We establish theoretical results confirming that the multi-scale prediction set retains the marginal coverage guarantees of the original conformal framework and can, in fact, yield smaller or more precise sets in practice. By distributing the total miscoverage probability across scales in proportion to their informative power, the method further refines the set sizes. We also show that dependence between scales can lead to conservative coverage, ensuring that the actual coverage exceeds the nominal level. Numerical experiments in a synthetic classification setting demonstrate that multi-scale conformal prediction achieves or surpasses the nominal coverage level while generating smaller prediction sets compared to single-scale conformal methods.