  There exist numerous proofs of Bell's theorem, stating that quantum mechanics
is incompatible with local realistic theories of nature. Here we define the
strength of such nonlocality proofs in terms of the amount of evidence against
local realism provided by the corresponding experiments. This measure tells us
how many trials of the experiment we should perform in order to observe a
substantial violation of local realism. Statistical considerations show that
the amount of evidence should be measured by the Kullback-Leibler or relative
entropy divergence between the probability distributions over the measurement
outcomes that the respective theories predict. The statistical strength of a
nonlocality proof is thus determined by the experimental implementation of it
that maximizes the Kullback-Leibler divergence from experimental (quantum
mechanical) truth to the set of all possible local theories. An implementation
includes a specification with which probabilities the different measurement
settings are sampled, and hence the maximization is done over all such setting
distributions.
  We analyze two versions of Bell's nonlocality proof (his original proof and
an optimized version by Peres), and proofs by Clauser-Horne-Shimony-Holt,
Hardy, Mermin, and Greenberger-Horne-Zeilinger. We find that the GHZ proof is
at least four and a half times stronger than all other proofs, while of the
two-party proofs, the one of CHSH is the strongest.
