  We perform a statistical inference of the astrophysical population of binary
black hole (BBH) mergers observed during the first two observing runs of
Advanced LIGO and Advanced Virgo, including events reported in the GWTC-1 and
IAS catalogs. We derive a novel formalism to fully and consistently account for
events of arbitrary significance. We carry out a software injection campaign to
obtain a set of mock astrophysical events subject to our selection effects, and
use the search background to compute the astrophysical probabilities $p_{\rm
astro}$ of candidate events for several phenomenological models of the BBH
population. We emphasize that the values of $p_{\rm astro}$ depend on both the
astrophysical and background models. Finally, we combine the information from
individual events to infer the rate, spin, mass, mass-ratio and redshift
distributions of the mergers. The existing population does not discriminate
between random spins with a spread in the effective spin parameter, and a small
but nonzero fraction of events from tidally-torqued stellar progenitors. The
mass distribution is consistent with one having a cutoff at $m_{\rm max} =
41^{+10}_{-5}\,\rm M_\odot$, while the mass ratio favors equal masses; the mean
mass ratio $\bar q> 0.67$. The rate shows no significant evolution with
redshift. We show that the merger rate restricted to BBHs with a primary mass
between 20 and $30\, \rm M_\odot$, and a mass ratio $q > 0.5$, and at $z \sim
0.2$, is 1.5 to $5.3\,{\rm Gpc^{-3} yr^{-1}}$ (90\% c.l.); these bounds are
model independent and a factor of $\sim 3$ tighter than that on the local rate
of all BBH mergers, and hence are a robust constraint on all progenitor models.
Including the events in our catalog increases the Fisher information about the
BBH population by $\sim 47\%$, and tightens the constraints on population
parameters.
