The Exact Hypergeometric Posterior Method for Accurate Inference of Population Size from Mark-Recapture Data

Reliable inference of census population size (N) from mark-recapture data is essential in ecology, conservation, and epidemiology, yet standard estimators often rely on asymptotic approximations that can misrepresent uncertainty. We introduce the Exact Hypergeometric Posterior (EHP), obtained by normalizing the hypergeometric likelihood for the classic two-sample design, which yields an exact posterior probability mass function over integer N. Closed-form normalization enables rapid evaluation of posterior summaries and exact highest-posterior-density (HPD) credible intervals. In this study, we show how heavy right tails naturally arise in sparse-recapture regimes and provide a principled truncation using an upper bound K, such as a plausible carrying capacity or sampling-frame limit, to regularize inference when needed. For repeated sampling, independent events can be combined by posterior multiplication and renormalization, yielding substantial gains in precision without large-sample approximations. Additionally, we extend the likelihood to account for loss of marked individuals and differential catchability by introducing an availability (retention) term $\varphi \left(t\right)$ and a relative catchability parameter $\omega$ via Fisher's noncentral hypergeometric model. Simulations and empirical examples confirm close agreement between analytic posteriors and Monte Carlo reference posteriors using our method. Together, these results provide a transparent, computationally tractable framework for finite-sample population-size inference in closed and partially open systems.