Figure: The “e-Lifting” Procedure for Combining Evidence Across Filtrations

Figure: Comparison of Anytime-Validity between Two E-Processes for Exchangeability Tests

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.

Citation

Choe, Y. J., & Ramdas, A. (2024). Combining Evidence Across Filtrations. arXiv preprint arXiv:2402.09698.

@article{choe2024combining,
  title={Combining Evidence Across Filtrations},
  author={Choe, Yo Joong and Ramdas, Aaditya},
  journal={arXiv preprint arXiv:2402.09698},
  year={2024}
}