In anytime-valid sequential inference, it is known that any admissible inference procedure must be based on test martingales and their composite generalization, called e-processes, which are nonnegative processes whose expectation at any arbitrary stopping time is upper-bounded by one. An e-process quantifies the accumulated evidence against a composite null hypothesis over a sequence of outcomes. This paper studies methods for combining e-processes that are computed using different information sets, i.e., filtrations, for a null hypothesis. Even though e-processes constructed on the same filtration can be combined effortlessly (e.g., by averaging), e-processes constructed on different filtrations cannot be combined as easily because their validity in a coarser filtration does not translate to validity in a finer filtration. We discuss three concrete examples of such e-processes in the literature: exchangeability tests, independence tests, and tests for evaluating and comparing forecasts with lags. Our main result establishes that these e-processes can be lifted into any finer filtration using adjusters, which are functions that allow betting on the running maximum of the accumulated wealth (thereby insuring against the loss of evidence). We also develop randomized adjusters that can improve the power of the resulting sequential inference procedure.

翻译：在任意有效序贯推断中，已知任何可容许的推断过程都必须基于检验鞅及其复合推广——称为e过程，即非负随机过程，其在任意停时的期望值均小于等于一。e过程量化了针对复合零假设在序列结果中积累的证据。本文研究针对同一零假设，利用不同信息集（即滤流）计算所得的e过程联合方法。尽管基于相同滤流构建的e过程可以轻松联合（例如通过取平均），但基于不同滤流构建的e过程却难以直接联合，因为其在较粗滤流中的有效性无法推广至更细滤流。我们讨论了文献中此类e过程的三个具体案例：可交换性检验、独立性检验，以及用于评估和比较含滞后预测的检验。我们的主要结论表明，通过使用调节函数——允许对累积财富的当前最大值进行投注（从而对冲证据损失）——可将这些e过程提升至任意更细滤流中。我们还开发了能够提升序贯推断过程统计功效的随机化调节函数。