  We use population inference to explore the impact that uncertainties in the
distribution of binary black holes (BBH) have on the astrophysical
gravitational-wave background (AGWB). Our results show that the AGWB monopole
is sensitive to the nature of the BBH population (particularly the local merger
rate), while the anisotropic $C_\ell$ spectrum is only modified to within a few
percent, at a level which is insignificant compared to other sources of
uncertainty (such as cosmic variance). This is very promising news for future
observational studies of the AGWB, as it shows that (i) the monopole can be
used as a new probe of the population of compact objects throughout cosmic
history, complementary to direct observations by LIGO and Virgo and (ii) we are
able to make surprisingly robust predictions for the $C_\ell$ spectrum, even
with only very approximate knowledge of the black hole population. As a result,
the AGWB anisotropies have enormous potential as a new probe of the large-scale
structure of the Universe, and of late-Universe cosmology in general.
