The role of multiple images and model priors in measuring $H_0$ from
  supernova Refsdal in galaxy cluster MACS J1149.5+2223
  Multiple image gravitational lensing systems with measured time delays
provide a promising one-step method for determining $H_0$. MACS J1149, which
lenses SN Refsdal into a quad S1-S4, and two other widely separated images, SX
and SY, is a perfect candidate. If time delays are pinned down, the remaining
uncertainty arises from the mass distribution in the lens. In MACS J1149, the
mass in the relevant lens plane region can be constrained by (i) many multiple
images, (ii) the mass of the galaxy splitting S1-S4 (which, we show, is
correlated with $H_0$), (iii) magnification of SX (also correlated with $H_0$),
and (iv) prior assumptions on the mass distribution. Our goal is not to
estimate $H_0$, but to understand its error budget, i.e., estimate
uncertainties associated with each of these constraints. Using multiple image
positions alone, yields very large uncertainty, despite the fact that the
position of SX is recovered to within $\!\le\!0.036$" (rms $\!\le\!0.36$") by
GRALE lens inversion. Fixing the mass of the galaxy that splits S1-S4 reduces
$1\sigma$ uncertainties to $\sim 23\%$, while fixing the magnification of SX
yields $1\sigma$ uncertainties of $32\%$. We conclude that smaller
uncertainties, of order few percent, are a consequence of imposing prior
assumptions on the shapes of the galaxy and cluster mass distributions, which
may or may not apply in a highly non-equilibrium environment of a merging
cluster. We propose that if a measurement of $H_0$ is to be considered
reliable, it must be supported by a wide range of lens inversion methods.
