  The relation between magnetic field strength B and gas density n in the
interstellar medium is of fundamental importance to many areas of astrophysics,
from protostellar disks to galaxy evolution. We present and compare Bayesian
analyses of the B - n relation for a comprehensive observational data set, as
well as a large body of numerical MHD simulations. We extend the original
Zeeman relation of Crutcher et al. (2010) with a large body of magnetic data
that includes 700 observations with the Davis-Chandrasekhar-Fermi method. By
using a new multiparameter Bayesian analysis we present a new, more general,
time-averaged observational relation: B \propto n^{0.27 \pm 0.017} for n \leq
n_0 and B \propto n^{0.54 \pm 0.18} for n \geq n_0, with n_0 = 924^(+145-144)
cm^-3. We perform a separate analysis on 19 numerical magnetohydrodynamics
simulations that cover a wide range of scales, resolutions, initial conditions,
and completed with a variety of codes: arepo, flash, pencil, and ramses. The
power law exponents derived from the simulations depend on several physical
factors including: dynamo effects, time scales, turbulence, and the initial
seed field strength. In particular, early-time simulations where the density,
velocity and magnetic fields are unevolved do not match the observational
scalings. The simulations that trace the observed density range best, the
evolved dwarf galaxy and Milky Way like galaxy simulations, settle into a near
consistent exponent of = 0.5 in the dense gas, with variability in the diffuse
gas exponent.
