  We describe a Bayesian formalism for analyzing individual gravitational-wave
events in light of the rest of an observed population. This analysis reveals
how the idea of a "population-informed prior" arises naturally from a suitable
marginalization of an underlying hierarchical Bayesian model which consistently
accounts for selection effects. Our formalism naturally leads to the presence
of "leave-one-out" distributions which include subsets of events. This differs
from other approximations, also known as empirical Bayes methods, which
effectively double count one or more events. We design a double-reweighting
post-processing strategy that uses only existing data products to reconstruct
the resulting population-informed posterior distributions. Although the
correction we highlight is an important conceptual point, we find it has a
limited impact on the current catalog of gravitational-wave events. Our
approach further allows us to study, for the first time in the
gravitational-wave literature, correlations between the parameters of
individual events and those of the population.
