  The current measurements of the cosmic ray energy spectrum at ultra-high
energies ($\text{E}>10^{19}$ eV) are characterized by large systematic errors
and poor statistics. In addition, the experimental results of the two
experiments with the largest published data sets, AGASA and HiRes, appear to be
inconsistent with each other, with AGASA seeing an unabated continuation of the
energy spectrum even at energies beyond the GZK cutoff energy at $10^{19.6}$
eV. Given the importance of the related astrophysical questions regarding the
unknown origin of these highly energetic particles, it is crucial that the
extent to which these measurements disagree be well understood. Here we
evaluate the consistency of the two measurements for the first time with a
model-independent method that accounts for the large statistical and systematic
errors of current measurements. We further compare the AGASA and HiRes spectra
with the recently presented Auger spectrum. The method directly compares two
measurements, bypassing the introduction of theoretical models for the shape of
the energy spectrum. The inconsistency between the observations is expressed in
terms of a Bayes Factor, a standard statistic defined as the ratio of a
separate parent source hypothesis to a single parent source hypothesis.
Application to the data shows that the two-parent hypothesis is disfavored. We
expand the method to allow comparisons between an experimental flux and that
predicted by any model.
