Weighted Asymptotically Optimal Sequential Testing

2025-11-10

AI Summary

This paper addresses the problem of effectively incorporating prior information into sequential multiple testing while rigorously preserving asymptotic optimality. To this end, we propose the Weighted Log-Likelihood Ratio (WLLR) framework and develop two novel procedures: the WLLR sequential test and the Weighted Gap-Intersection procedure. Our work introduces, for the first time in sequential multiple testing, a provably first-order optimal weighting mechanism. We establish theoretical guarantees showing that the expected stopping time achieves the information-theoretic lower bound, and that the family-wise error rate (FWER) is strongly controlled. The proposed methods are robust under high-dimensional settings, random weight assignments, and heterogeneous signal strengths—overcoming the longstanding limitation of conventional sequential tests, which struggle to simultaneously leverage prior knowledge and maintain theoretical optimality.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Information Theory
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.

Problem

• Incorporating prior information into sequential multiple testing procedures
• Maintaining asymptotic optimality with weighted log-likelihood ratio modifications
• Achieving theoretical lower bound for expected stopping times

Innovation

• Weighted log-likelihood ratio modifies standard LLR
• Weighted Gap procedures control family-wise error rate
• Asymptotically optimal stopping times achieve theoretical lower bound