The Hidden Prior: Variance Constraints Under Data Augmentation
Data augmentation is now a standard device across capture--recapture and occupancy analysis: adding a fixed number M of all-zero encounter histories replaces a model of unknown dimension with one of fixed dimension. Although M is often treated as a computational tuning choice, it also specifies a finite superpopulation and hence a binomial support constraint on the number of undetected individuals. In a Bayesian implementation that constraint appears as an induced prior; in a likelihood implementation it is the same finite-support assumption reached by another route. That the Bernoulli specification for the inclusion indicators induces a binomial prior on abundance is established (Schofield & Barker 2014); our concern is what that choice costs in estimated uncertainty. We develop the argument using a simple closed-population abundance estimation problem. We show that augmented occupancy and Huggins conditional-likelihood analyses give numerically identical point estimates of N once M is sufficiently large. Their uncertainty estimates, however, need not agree. We distinguish two sources of discrepancy. First, when M is small relative to the number of undetected individuals, the finite binomial ceiling truncates the likelihood or posterior and suppresses uncertainty. Second, once that ceiling no longer binds, Taylor-series (Delta-method) approximations still understate variance, because the quantity of interest is a strongly non-linear function of the estimated parameters and local linearization does not reproduce its curvature. Gauss-Hermite quadrature on the unconstrained logit scale recovers much of the shortfall and approaches the MCMC posterior benchmark, though a small residual remains that does not close as M grows, reflecting the distinction between asymptotic likelihood theory and finite-sample Bayesian inference. Neither mechanism is peculiar to abundance estimation: the first follows from the augmented representation itself, the second from any derived quantity that is a non-linear function of estimated parameters. We close with framework-specific guidance for choosing M.