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

Area of Science:

• Ecology
• Conservation Biology
• Epidemiology

Background:

• Standard mark-recapture estimators for population size (N) often use approximations, leading to inaccurate uncertainty estimates.
• Accurate population size inference is critical for ecological, conservation, and epidemiological applications.

Purpose of the Study:

• Introduce the Exact Hypergeometric Posterior (EHP) for exact, finite-sample population size inference.
• Provide a computationally tractable framework for ecological and epidemiological modeling.

Main Methods:

• Derived the EHP by normalizing the hypergeometric likelihood for a two-sample mark-recapture design.
• Developed closed-form solutions for posterior summaries and exact highest-posterior-density (HPD) credible intervals.
• Incorporated principled truncation for sparse data and extended the model for individual loss and catchability variation using Fisher's noncentral hypergeometric model.

Main Results:

• The EHP provides an exact posterior probability mass function for integer N.
• Demonstrated natural emergence of heavy right tails in sparse-recapture data and introduced regularization via an upper bound K.
• Showcased methods for combining repeated sampling via posterior multiplication and renormalization for enhanced precision.
• Validated analytic posteriors against Monte Carlo methods in simulations and empirical examples.

Conclusions:

• The EHP offers a transparent and computationally efficient framework for population size estimation.
• The method provides exact inference, overcoming limitations of asymptotic approximations in mark-recapture studies.
• The framework is applicable to both closed and partially open ecological systems.