Combining Evidence Across Filtrations

2024-02-15

arXiv.org

AI Summary

In sequential anytime testing, combining e-process evidence across filtrations poses a fundamental challenge: an e-process valid under a coarse filtration may fail to retain validity under a finer one, even for the same null hypothesis. Method: We propose the “adjusted combination” framework, introducing the class of *adjuster* functions—characterizing necessary and sufficient conditions for cross-filtration evidence boosting. We prove that an adjuster is essential to restore anytime validity under the original filtration and quantify its logarithmic cost. Our approach unifies e-process theory, generalized test martingales, and filtration coarsening/refinement techniques. Results: We establish a complete characterization theorem for adjusters and validate the framework on real financial data for randomness testing. The work provides novel theoretical tools and practical methodology for sequential independence testing and predictive model evaluation.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

In sequential anytime-valid inference, any admissible procedure must be based on e-processes: generalizations of test martingales that quantify the accumulated evidence against a composite null hypothesis at any stopping time. This paper proposes a method for combining e-processes constructed in different filtrations but for the same null. Although e-processes in the same filtration can be combined effortlessly (by averaging), e-processes in different filtrations cannot because their validity in a coarser filtration does not translate to a finer filtration. This issue arises in sequential tests of randomness and independence, as well as in the evaluation of sequential forecasters. We establish that a class of functions called adjusters can lift arbitrary e-processes across filtrations. The result yields a generally applicable"adjust-then-combine"procedure, which we demonstrate on the problem of testing randomness in real-world financial data. Furthermore, we prove a characterization theorem for adjusters that formalizes a sense in which using adjusters is necessary. There are two major implications. First, if we have a powerful e-process in a coarsened filtration, then we readily have a powerful e-process in the original filtration. Second, when we coarsen the filtration to construct an e-process, there is a logarithmic cost to recovering validity in the original filtration.

Problem

• Combining e-processes across different filtrations
• Testing randomness and independence sequentially
• Lifting e-processes using adjusters

Innovation

• Combining e-processes across filtrations
• Using adjusters for lifting e-processes
• Testing randomness in financial data