Carefree multiple testing with e-processes

E-processes enable hypothesis testing with ongoing data collection while maintaining Type I error control. However, when testing multiple hypotheses simultaneously, current e-value based multiple testing methods such as e-BH are not invariant to the order in which data are gathered for the different e-processes. This can lead to undesirable situations, e.g., where a hypothesis rejected at time t is no longer rejected at time t+1 after choosing to gather more data for one or more e-processes unrelated to that hypothesis. We argue that multiple testing methods should always work with suprema of e-processes. We provide an example to illustrate that e-BH does not control the FDR, at level α when applied to suprema of e-processes. From the same example we see that the FWER is not controlled with averaging, and also closed e-BH does not control the FDR. We show that adjusters can be used to ensure FDR-sup control with e-BH under arbitrary dependence.