  In the local universe, the masses of Super-Massive Black-Holes (SMBH) appear
to correlate with the physical properties of their hosts, including the mass of
the dark-matter halo. Using these clues as a starting point many studies have
produced models that can explain phenomena like the quasar luminosity function.
The shortcoming of this approach is that working models are not unique, and as
a result it is not always clear what input physics is being constrained. Here
we take a different approach. We identify critical parameters that describe the
evolution of SMBHs at high redshift, and constrain their parameter space based
on observations of high redshift quasars from the Sloan Digital Sky Survey. We
find that the luminosity function taken in isolation is somewhat limited in its
ability to constrain SMBH evolution due to some strong degeneracies. This
explains the presence in the literature of a range of equally successful models
based on different physical hypotheses. Including the constraint of the local
SMBH to halo mass ratio breaks some of the degeneracies, and our results
suggest halo masses at z~4.8 of 10^{12.5+/-0.3}M_solar (with 90% confidence),
with a SMBH to halo mass ratio that decreases with time (>99%). We also find a
quasar luminosity to halo mass ratio that increases with halo mass (>99%).
These features need to be incorporated in all successful models of SMBH
evolution. On the other hand current observations do not permit any conclusions
regarding the evolution of quasar lifetime, or the SMBH occupation fraction in
dark matter halos.
