We explore scenarios for reionizing the intergalactic medium with low galaxy ionizing photon escape fractions . We combine simulation-based halo-mass dependent escape fractions with an extrapolation of the observed galaxy rest-ultraviolet luminosity functions to solve for the reionization history from z = 20 \rightarrow 4 . We explore the posterior distributions for key unknown quantities , including the limiting halo mass for star-formation , the ionizing photon production efficiency , and a potential contribution from active galactic nuclei ( AGN ) . We marginalize over the allowable parameter space using a Markov Chain Monte Carlo method , finding a solution which satisfies the most model-independent constraints on reionization . Our fiducial model can match observational constraints with an average escape fraction of < 5 % throughout the bulk of the epoch of reionization if : i ) galaxies form stars down to the atomic cooling limit before reionization and a photosuppression mass of log ( M _ { \mathrm { h } } /M _ { \odot } ) \sim 9 during/after reionization ( - 13 < M _ { \mathrm { UV,lim } } < - 11 ) ; ii ) galaxies become more efficient producers of ionizing photons at higher redshifts and fainter magnitudes , and iii ) there is a significant , but sub-dominant , contribution by AGN at z \lesssim 7 . In this model the faintest galaxies ( M _ { \mathrm { UV } } > - 15 ) dominate the ionizing emissivity , leading to an earlier start to reionization and a smoother evolution of the ionized volume filling fraction than models which assume a single escape fraction at all redshifts and luminosities . The ionizing emissivity from this model is consistent with observations at z = 4–5 ( and below , when extrapolated ) , in contrast to some models which assume a single escape fraction . Our predicted ionized volume filling fraction at z = 7 of Q _ { H _ { II } } = 78 % ( \pm 8 % ) is in modest ( \sim 1–2 \sigma ) tension with observations of Ly \alpha emitters at z \sim 7 and the damping wing analyses of the two known z > 7 quasars , which prefer Q _ { H _ { II } ,z = 7 } \sim 40–50 % .