Posterior Top-K Selection under Noisy Reviews: Equivalence and Gain Relative to Posterior Mean

Selection problems, including conference acceptance, grant review, and hiring, often require choosing exactly $K$ candidates from noisy and heterogeneous evaluations. Ranking aggregate scores does not directly answer the decision-relevant question: how likely is each candidate to belong to the latent top $K$? We formalize this target through the posterior membership probability $p_i=\mathrm{Pr}(i\in \mathrm{TopK}(q)\mid D)$. Selecting the $K$ largest $p_i$ is Bayes-optimal for expected overlap with the latent top-$K$ set. Taking this established decision rule as a starting point, we study when it differs from posterior-mean ranking, why it differs, and how much the resulting roster change matters. We show that exchangeable centered residuals yield a structural equivalence regime. For independent posteriors with continuous densities, we derive a boundary approximation under local regularity; its Gaussian case predicts uncertainty-driven reordering. For arbitrary joint posteriors, we further show that the consequence of disagreement depends jointly on how many candidates change and how far those candidates lie from the membership-probability boundary, and establish when this characterization stabilizes in growing populations. Simulations illustrate these theoretical predictions. Our analysis of ICLR 2026 reviews finds identical rosters at the realized cohort capacity, consistent with nearly homogeneous posterior uncertainty across papers. The Jester results show occasional roster differences, but these yield only small estimated gains under the fitted posterior, while evaluation against an independent high-information reference does not establish a clear advantage. Overall, roster changes alone do not determine the value of posterior Top-$K$ relative to posterior-mean ranking.