  Observations of gravitational waves emitted by merging compact binaries have
provided tantalising hints about stellar astrophysics, cosmology, and
fundamental physics. However, the physical parameters describing the systems,
(mass, spin, distance) used to extract these inferences about the Universe are
subject to large uncertainties. The most widely-used method of performing these
analyses requires performing many Monte Carlo integrals to marginalise over the
uncertainty in the properties of the individual binaries and the survey
selection bias. These Monte Carlo integrals are subject to fundamental
statistical uncertainties. Previous treatments of this statistical uncertainty
has focused on ensuring the precision of the inferred inference is unaffected,
however, these works have neglected the question of whether sufficient accuracy
can also be achieved. In this work, we provide a practical exploration of the
impact of uncertainty in our analyses and provide a suggested framework for
verifying that astrophysical inferences made with the gravitational-wave
transient catalogue are accurate. Applying our framework to models used by the
LIGO-Virgo-KAGRA collaboration and in the wider literature, we find that Monte
Carlo uncertainty in estimating the survey selection bias is the limiting
factor in our ability to probe narrow population models and this will rapidly
grow more problematic as the size of the observed population increases.
