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.
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