G Additional simulations for the two-samples prob- lem Table 6 shows a comparison of the performance of the minimum p -value and Fisher- combined test statistics. The variant with highest power in most setups (specifically, setups 2,4,5,6,8,9,10,11,12) is the Fisher-combined p -value using aggregation by sum- mation and m$_{max}$ = 50. However, the Fisher-combined p -value test is sensitive to the choice of m$_{max}$ , which is unknown in practice. Since we view this as a significant weakness of the Fisher-combined statistic, and since the minimum p -value does not have this weakness and has very good power when compared with Fisher as well as when compared with other tests (with a large range of m$_{max}$ values examined), we recommend the minimum p -value test statistic. Table 7 shows the power using different priors for regularization. The priors are as follows: π ( m ) = √ N $^{m}$exp ( − √ N ) /m ! for Poisson; π ( m ) = ( N − 1 m − 1 $^{)}$p $^{m}$(1 − p ) ( N − m ) for Table 6: Power of the minimum p -value test statistic (columns 2–5) as well as the Fisher-combined p -value test statistic (columns 6–9), using the different aggregation methods and two values of m$_{max}$ , for N = 500 and the setups of Figure 2. The difference between m$_{max}$ = 50 and m$_{max}$ = 149 is much larger for the Fisher-combined than for the minimum p -value test statistic when aggregated by summation (columns 8–9 versus columns 4–5). Moreover, the power is typically lower for the Fisher- combined that for the minimum p -value test statistic when the aggregation is by maximization (columns 6–7 versus columns 2–3). 66 Minimum p -value Fisher combined p -value Max aggregation Sum aggregation Max aggregation Sum aggregation Setup m$_{max}$ = 50 m$_{max}$ = 149 m$_{max}$ = 50 m$_{max}$ = 149 m$_{max}$ = 50 m$_{max}$ = 149 m$_{max}$ = 50 m$_{max}$ = 149 1 0.836 0.825 0.500 0.491 0.631 0.424 0.549 0.560 2 0.810 0.799 0.883 0.873 0.819 0.701 0.917 0.898 3 0.783 0.785 0.553 0.733 0.867 0.847 0.502 0.782 4 0.836 0.827 0.945 0.937 0.749 0.567 0.952 0.882 5 0.607 0.592 0.708 0.686 0.531 0.381 0.760 0.655 6 0.829 0.818 0.836 0.820 0.749 0.567 0.882 0.821 7 0.358 0.339 0.516 0.492 0.201 0.135 0.430 0.287 8 0.770 0.752 0.795 0.775 0.542 0.361 0.814 0.672 9 0.531 0.512 0.641 0.613 0.344 0.227 0.635 0.470 10 0.728 0.711 0.824 0.806 0.569 0.393 0.849 0.721 11 0.556 0.540 0.707 0.686 0.509 0.380 0.756 0.657 12 0.402 0.390 0.599 0.577 0.399 0.293 0.655 0.557 13 0.852 0.844 0.777 0.764 0.809 0.663 0.823 0.822