Optimized combination of independent or simultaneous e-values

2026-03-11

AI Summary

This study addresses the challenge of effectively combining e-values under data-dependent tuning while preserving statistical validity. The authors develop a class of optimized e-process–based combination methods tailored for both independent settings and a newly introduced notion of “simultaneous e-variables.” They establish, for the first time, that combined tests retain validity even when tuning parameters are selected in a data-driven manner. By incorporating elementary symmetric polynomials, the proposed approach enhances statistical power and establishes a flexible intermediate framework that bridges independent and sequential validity. Through rigorous theoretical analysis grounded in dependence structure modeling, the method achieves substantially improved detection capability while maintaining strict Type-I error control.

Technology Category

• Machine Learning: Ensemble Methods
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We show that a class of optimized e-value combinations, arising from a standard construction of e-processes, remains valid even when the tuning parameter is optimized based on the data. This result holds for independent e-values, and, more generally, for a new class called simultaneous e-variables, whose dependence structure lies between independence and sequential validity. We further propose an improved combination test for such e-values based on elementary symmetric polynomials.

Problem

• e-values
• data-dependent tuning
• statistical validity
• simultaneous e-variables
• combination tests

Innovation

• e-values
• simultaneous e-variables
• e-processes
• elementary symmetric polynomials
• data-dependent tuning