In contemporary research, data scientists often test an infinite sequence of hypotheses $H_1,H_2,\ldots$ one by one, and are required to make real-time decisions without knowing the future hypotheses or data. In this paper, we consider such an online multiple testing problem with the goal of providing simultaneous lower bounds for the number of true discoveries in data-adaptively chosen rejection sets. Employing the recent online closure principle, we show that for this task it is necessary to use an anytime-valid test for each intersection hypothesis. This connects two distinct branches of the literature: online testing of multiple hypotheses (where the hypotheses appear online), and sequential anytime-valid testing of a single hypothesis (where the data for a fixed hypothesis appears online). Motivated by this result, we construct a new online closed testing procedure and a corresponding short-cut with a true discovery guarantee based on multiplying sequential e-values. This general but simple procedure gives uniform improvements over the state-of-the-art methods but also allows to construct entirely new and powerful procedures.

翻译：在当代研究中，数据科学家经常逐一检验无限序列的假设$H_1,H_2,\\ldots$，并需在未知未来假设或数据的情况下做出实时决策。本文针对此类在线多重检验问题，旨在为数据自适应选择的拒绝集合中真实发现的数量提供同步下界。通过应用最新的在线闭包原理，我们证明为此任务必须对每个交集假设使用任意时间有效的检验方法。这连接了文献中两个独立分支：多重假设的在线检验（假设在线出现）与单一假设的序列化任意时间有效检验（固定假设的数据在线出现）。基于此结果，我们构建了一种新的在线闭包检验流程及其对应捷径，该方案通过累乘序列e值实现真实发现保证。这一通用而简洁的流程不仅对现有最优方法实现了均匀改进，还能构建全新且高效的检验程序。