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.
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