TDCOSMO IV: Hierarchical time-delay cosmography -- joint inference of
  the Hubble constant and galaxy density profiles
  The H0LiCOW collaboration inferred via gravitational lensing time delays a
Hubble constant $H_0=73.3^{+1.7}_{-1.8}$ km s$^{-1}{\rm Mpc}^{-1}$, describing
deflector mass density profiles by either a power-law or stars plus standard
dark matter halos. The mass-sheet transform (MST) that leaves the lensing
observables unchanged is considered the dominant source of residual uncertainty
in $H_0$. We quantify any potential effect of the MST with flexible mass models
that are maximally degenerate with H0. Our calculation is based on a new
hierarchical approach in which the MST is only constrained by stellar
kinematics. The approach is validated on hydrodynamically simulated lenses. We
apply the method to the TDCOSMO sample of 7 lenses (6 from H0LiCOW) and measure
$H_0=74.5^{+5.6}_{-6.1}$ km s$^{-1}{\rm Mpc}^{-1}$. In order to further
constrain the deflector mass profiles, we then add imaging and spectroscopy for
33 strong gravitational lenses from the SLACS sample. For 9 of the SLAC lenses
we use resolved kinematics to constrain the stellar anisotropy. From the joint
analysis of the TDCOSMO+SLACS sample, we measure $H_0=67.4^{+4.1}_{-3.2}$ km
s$^{-1}{\rm Mpc}^{-1}$, assuming that the TDCOSMO and SLACS galaxies are drawn
from the same parent population. The blind H0LiCOW, TDCOSMO-only and
TDCOSMO+SLACS analyses are in mutual statistical agreement. The TDCOSMO+SLACS
analysis prefers marginally shallower mass profiles than H0LiCOW or
TDCOSMO-only. While our new analysis does not statistically invalidate the mass
profile assumptions by H0LiCOW, and thus their $H_0$ measurement relying on
those, it demonstrates the importance of understanding the mass density profile
of elliptical galaxies. The uncertainties on $H_0$ derived in this paper can be
reduced by physical or observational priors on the form of the mass profile, or
by additional data, chiefly spatially resolved kinematics of lens galaxies.
