Combining Evidence Across Filtrations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910053081022464 |
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| author | Choe, Yo Joong Ramdas, Aaditya |
| author_facet | Choe, Yo Joong Ramdas, Aaditya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09698 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combining Evidence Across Filtrations Choe, Yo Joong Ramdas, Aaditya Methodology Machine Learning Probability Statistics Theory 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. |
| title | Combining Evidence Across Filtrations |
| topic | Methodology Machine Learning Probability Statistics Theory |
| url | https://arxiv.org/abs/2402.09698 |