  The neutron star equation of state is now being constrained from a diverse
set of multi-messenger data, including gravitational waves from binary neutron
star mergers, X-ray observations of the neutron star radius, and many types of
laboratory nuclear experiments. These measurements are often mapped to a common
domain for comparison with one another or are used to constrain the predictions
of theoretical equations of state. We explore here the statistical biases that
can arise when such multi-messenger data are compared or combined across
different domains. We find that placing Bayesian priors individually in each
domain of measurement can lead to biased constraints. We present a new
prescription for defining Bayesian priors consistently across different
experiments, which will allow for robust cross-domain comparisons. Using the
first two binary neutron star mergers as an example, we show that a uniform
prior in the tidal deformability can produce inflated evidence for large radii,
while a uniform prior in the radius points towards smaller radii. Finally,
using this new prescription, we provide a status update on multi-messenger
constraints on the neutron star radius.
