Data-light Uncertainty Set Merging with Admissibility: Synthetics, Aggregation, and Test Inversion

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 control levels, and it adapts to any user-specified initial uncertainty sets, accommodating potentially varying coverage levels. 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 a proof of 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.