Combining Evidence Across Filtrations

TL;DR

This paper introduces a method to combine e-processes across different filtrations in sequential inference, enabling more flexible and powerful testing procedures, especially in financial data analysis.

Contribution

The paper proposes adjusters that lift e-processes across filtrations, establishing an 'adjust-then-combine' method with theoretical characterization and practical application.

Findings

Effective combination of e-processes across filtrations.

Logarithmic cost in validity recovery when coarsening filtrations.

Demonstration on testing randomness in financial data.

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.