where x is the GW and AGN data . We consider the AGN association ( i.e. λ > 0 , because at least a fraction of the BBH come from AGNs ) to be confident if K > 100 . The number of follow - up observations needed to reach this requirement as a function of the true value of λ given in input is shown in Fig .   3 for different sky maps . It is clear that for poorly localized events like GW190521 , tens of GW follow up campaigns are Figure 3 . Number of GW events to be followed - up to obtain a confident association ( i.e. for a Bayes factor K = 100 ) between AGN flares and BBH events , as a function of the true input value of λ in the simulations . Each curve corresponds to a different fixed GW sky map , with a range of localization volumes ( from ∼ 10 −4 to 10 Gpc 3 at 99 % CI ) . This figure applies to the ideal case in which the AGN flares associated to the GW events are all detectable ( we assume a magnitude limit g < 20.5 ) . The orange line shows a comparison with the result from Bartos et al . ( 2017a ) ( B17 ) , who do not consider follow - up observations of AGN flares , but instead compare the GW localizations with AGN positions . The other differences with B17 are that their N corresponds to the number of events needed to reach 3σ using a p−value statistical method , and that they consider a slightly lower AGN number density . required in order to make a confident association , even in the most optimistic case where λ tr = 1 and we can detect all AGNs where the BBH merger could happen . Only a few to tens of events are needed to make a confident association for better - localized events such as GW170814 or GW190814 down to λ tr = 0.1 . This is an obvious consequence of the fact that the number of background contaminants scales with the comoving volume , and that the localization volume of these events is orders of magnitude lower than the one of GW190521 . We compare our results with the predictions by B17 who uses only GW localizations ( i.e. without follow - up observations ) to probe the origin of BBH mergers and to potentially associate them with AGNs . Fig . 3 shows that we find a similar scaling relation of the number of events required to reach K = 100 as a function of λ as theirs , although slightly less steep than N(K = 100 ) ∝ λ −2 . The main differences with the B17 predictions , other than follow - up observations , are that their N corresponds to the number of events needed to reach 3σ using a p−value statistical method ( so that here the scaling of the number of events needed could be different as we are not directly considering the width of a distribution ) , and that they consider a fixed ( in redshift ) and lower AGN number density .