FIG. 4: Coverage probability of Method-1 posterior credibility upper limits (left) and central intervals (right), as a function of the number of experiment replications N$_{R}$ . The solid lines indicate the nominal credibility. FIG. 5: Variation of the Method 1 reference posterior upper limit with mean background for several values of the observed number of events n . The relative uncertainty on the background and on the effective luminosity is 20% for the left plot and 50% for the right one. that upper limits will not be exactly constant. The n = 0 case is illustrated in Fig. 5 for two values of the relative uncertainties on background and signal efficiency. For n > 0 the likelihood function still factorizes approximately since ( µ + ϵσ ) n ≈ µ $^{n}$(1 + nϵσ/µ ) ≈ µ n for µ ≫ ϵσ, n . Thus upper limits will flatten out at large ¯ µ , as seen in Fig. 5. A comparison 22