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.

翻译：在序列性随时有效推断中，任何可容许的程序都必须基于e过程：这是对检验鞅的推广，用于量化在任意停止时间下针对复合零假设所累积的证据。本文提出了一种在不同过滤过程中但针对相同零假设构建的e过程进行组合的方法。尽管同一过滤过程中的e过程可以轻松组合（通过平均），但不同过滤过程中的e过程则无法直接组合，因为它们在较粗过滤中的有效性无法直接转化到较细的过滤中。这一问题出现在随机性与独立性的序列检验中，以及在序列预测器的评估中。我们证明了一类称为调整器的函数可以将任意e过程在不同过滤过程之间提升。该结果产生了一种普遍适用的“先调整后组合”程序，我们通过在真实世界金融数据中检验随机性的问题展示了这一方法。此外，我们证明了调整器的一个表征定理，该定理形式化了使用调整器在某种意义上是必要的这一观点。这带来了两个主要影响。首先，如果我们在一个粗化的过滤中拥有一个强力的e过程，那么我们就能轻易地在原始过滤中获得一个强力的e过程。其次，当我们粗化过滤以构建一个e过程时，要在原始过滤中恢复有效性会付出一个对数代价。