Admissible online closed testing must employ e-values

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.