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.

翻译：在任意时间有效的序贯推断中，已知任何可容许的推断程序都必须基于e-过程——这是检验鞅的复合推广，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。本文研究了为同一零假设在不同信息集（过滤）下构建的e-过程的融合方法。虽然在相同过滤下构建的e-过程可以轻松融合（例如通过取平均），但在不同过滤下构建的e-过程则无法直接融合，因为它们在较粗过滤下的有效性并不能转化为在较细过滤下的有效性。这一问题出现在可交换性检验、独立性检验以及带滞后项的预测比较检验中。我们首先证明一类称为调整器的函数能够将e-过程从较粗过滤提升至任意较细过滤。随后我们提出了调整器的表征定理，从形式化层面说明了使用调整器的必要性。这带来两个主要推论：第一，若在粗化过滤中存在强效的e-过程，则我们可立即获得原始过滤中的强效e-过程；第二，当我们通过粗化过滤构建e-过程时，在原始过滤中恢复任意时间有效性需付出渐近对数的代价。