# Combining Evidence Across Filtrations

arXiv: https://arxiv.org/abs/2402.09698 · HTML: https://mlsanity.com/paper/2402.09698/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 56 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 29 of 29)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Testing randomness](https://mlsanity.com/paper/1906.09256/) (2019)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Retrain or not retrain: Conformal test martingales for change-point detection](https://mlsanity.com/paper/2102.10439/) (2021)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [Sequential Nonparametric Testing with the Law of the Iterated Logarithm](https://mlsanity.com/paper/1506.03486/) (2015)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities](https://mlsanity.com/paper/2304.02611/) (2023)
- [Post-selection inference for e-value based confidence intervals](https://mlsanity.com/paper/2203.12572/) (2022)
- [Derandomized knockoffs: leveraging e-values for false discovery rate control](https://mlsanity.com/paper/2205.15461/) (2022)
- [Testing exchangeability by pairwise betting](https://mlsanity.com/paper/2310.14293/) (2023)
- [A unified framework for bandit multiple testing](https://mlsanity.com/paper/2107.07322/) (2021)
- [Comparing Sequential Forecasters](https://mlsanity.com/paper/2110.00115/) (2021)
- [E-Statistics, Group Invariance and Anytime Valid Testing](https://mlsanity.com/paper/2208.07610/) (2022)
- [Valid sequential inference on probability forecast performance](https://mlsanity.com/paper/2103.08402/) (2021)
- [Sequentially valid tests for forecast calibration](https://mlsanity.com/paper/2109.11761/) (2021)
- [Conformal testing in a binary model situation](https://mlsanity.com/paper/2104.01885/) (2021)
- [Multiple testing under negative dependence](https://mlsanity.com/paper/2212.09706/) (2022)
- [ALL-IN meta-analysis: breathing life into living systematic reviews and prospective meta-analyses](https://mlsanity.com/paper/2109.12141/) (2021)
- [E-values as unnormalized weights in multiple testing](https://mlsanity.com/paper/2204.12447/) (2022)
- [The extended Ville's inequality for nonintegrable nonnegative supermartingales](https://mlsanity.com/paper/2304.01163/) (2023)
- [A Rank-Based Sequential Test of Independence](https://mlsanity.com/paper/2305.13818/) (2023)

## Built on by (most cited first, 3 of 3)

- [Bayes with No Shame: Admissibility Geometries of Predictive Inference](https://mlsanity.com/paper/2603.05335/) (2026)
- [When the Martingale Never Stops Firing: Anytime-Valid Gating on Real Forecast Streams](https://mlsanity.com/paper/2608.30502/) (2026)
- [The Premise Is the Problem: Exchangeability Failure in Self-Monitored Test-Time Adaptation](https://mlsanity.com/paper/2610.07038/) (2026)

## Abstract (arXiv)

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.