Combining Evidence Across Filtrations Using Adjusters
Abstract
In anytime-valid sequential inference, it is known that any admissible procedure must be based on e-processes, which are composite generalizations of test martingales that quantify the accumulated evidence against a composite null hypothesis at any arbitrary stopping time. This paper studies methods for combining e-processes constructed using different information sets (filtrations) for the same null. Although e-processes constructed in the same filtration can be combined effortlessly (e.g., by averaging), e-processes constructed in different filtrations cannot, because their validity in a coarser filtration does not translate to validity in a finer filtration. This issue arises in exchangeability tests, independence tests, and tests for comparing forecasts with lags. We first establish that a class of functions called adjusters allows us to lift e-processes from a coarser filtration into any finer filtration. We then introduce a characterization theorem for adjusters, formalizing 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 an asymptotically logarithmic cost of recovering anytime-validity in the original filtration.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2024
- DOI:
- 10.48550/arXiv.2402.09698
- arXiv:
- arXiv:2402.09698
- Bibcode:
- 2024arXiv240209698C
- Keywords:
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- Statistics - Methodology;
- Computer Science - Machine Learning;
- Mathematics - Probability;
- Mathematics - Statistics Theory;
- Statistics - Machine Learning
- E-Print:
- Substantially revised with new results in Sections 5 and 6. Code is available at https://github.com/yjchoe/CombiningEvidenceAcrossFiltrations