Data-light Uncertainty Set Merging with Admissibility

TL;DR

This paper presents a data-light method called SAT for merging diverse uncertainty sets into a single set, ensuring reliable coverage even with dependent data, and introduces theoretical guarantees of admissibility.

Contribution

The paper introduces SAT, a novel synthetic test-based approach for merging uncertainty sets that is flexible, data-light, and theoretically admissible.

Findings

SAT guarantees finite-sample coverage validity.

The method produces smaller, reliable uncertainty sets.

Theoretical analysis confirms admissibility of the merging procedure.

Abstract

This article introduces a Synthetics, Aggregation, and Test inversion (SAT) approach for merging diverse and potentially dependent uncertainty sets into a single unified set. The procedure is data-light, relying only on initial sets and their nominal levels, and it flexibly adapts to user-specified input sets with possibly varying coverage guarantees. SAT is motivated by the challenge of integrating uncertainty sets when only the initial sets and their control levels are available-for example, when merging confidence sets from distributed sites under communication constraints or combining conformal prediction sets generated by different algorithms or data splits. To address this, SAT constructs and aggregates novel synthetic test statistics, and then derive merged sets through test inversion. Our method leverages the duality between set estimation and hypothesis testing, ensuring reliable coverage in dependent scenarios. A key theoretical contribution is a rigorous analysis of SAT's properties, including its admissibility in the context of deterministic set merging. Both theoretical analyses and empirical results confirm the method's finite-sample coverage validity and desirable set sizes.