  Using the most recent measurements of the ultraviolet (UV) luminosity
functions (LFs) and dust estimates of early galaxies, we derive updated
dust-corrected star-formation rate functions (SFRFs) at z~4-8, which we model
to predict the evolution to higher redshifts, z>8. We employ abundance matching
techniques to calibrate a relation between galaxy star formation rate (SFR) and
host halo mass M$_{h}$ by mapping the shape of the observed SFRFs at z~4-8 to
that of the halo mass function. The resulting scaling law remains roughly
constant over this redshift range. We apply the average SFR-M$_{h}$ relation to
reproduce the observed SFR functions at 4 <= z <= 8 and also derive the
expected UV LFs at higher redshifts. At z~9 and z~10 these model LFs are in
excellent agreement with current observed estimates. Our predicted number
densities and UV LFs at z>10 indicate that JWST will be able to detect galaxies
out to z~15 with an extensive treasury sized program. We also derive the
redshift evolution of the star formation rate density and associated
reionization history by galaxies. Models which integrate down to the current
HUDF12/XDF detection limit (M$_{UV}$ ~ -17.7 mag) result in a SFRD that
declines as (1+z)$^{-10.4\pm0.3}$ at high redshift and fail to reproduce the
observed CMB electron scattering optical depth, $\tau$ ~ 0.066, to within
1$\sigma$. On the other hand, we find that the inclusion of galaxies with SFRs
well below the current detection limit (M$_{UV}$ < -5.7 mag) leads to a fully
reionized universe by z ~ 6.5 and an optical depth of $\tau$ ~ 0.054,
consistent with the recently derived Planck value at the 1$\sigma$ level.
