  The Cosmological Principle, that the Universe is homogeneous and isotropic on
sufficiently large scales, underpins the standard model of cosmology. However,
a recent analysis of 1.36 million infrared-selected quasars has identified a
significant tension in the amplitude of the number-count dipole compared to
that derived from the CMB, thus challenging the Cosmological Principle. Here we
present a Bayesian analysis of the same quasar sample, testing various
hypotheses using the Bayesian evidence. We find unambiguous evidence for the
presence of a dipole in the distribution of quasars with a direction that is
consistent with the dipole identified in the CMB. However, the amplitude of the
dipole is found to be 2.7 times larger than that expected from the conventional
kinematic explanation of the CMB dipole, with a statistical significance of
$5.7\sigma$. To compare these results with theoretical expectations, we sharpen
the $\Lambda$CDM predictions for the probability distribution of the amplitude,
taking into account a number of observational and theoretical systematics. In
particular, we show that the presence of the Galactic plane mask causes a
considerable loss of dipole signal due to a leakage of power into higher
multipoles, exacerbating the discrepancy in the amplitude. By contrast, we show
using probabilistic arguments that the source evolution of quasars improves the
discrepancy, but only mildly so. These results support the original findings of
an anomalously large quasar dipole, independent of the statistical methodology
used.
