Hierarchical modeling of abundance in closed population capture–recapture models under heterogeneity

Hierarchical modeling of abundance in space or time using closed-population mark-recapture under heterogeneity (model $$\hbox {M}_{\text {h}}$$Mh) presents two challenges: (i) finding a flexible likelihood in which abundance appears as an explicit parameter and (ii) fitting the hierarchical model for abundance. The first challenge arises because abundance not only indexes the population size, it also determines the dimension of the capture probabilities in heterogeneity models. A common approach is to use data augmentation to include these capture probabilities directly into the likelihood and fit the model using Bayesian inference via Markov chain Monte Carlo (MCMC). Two such examples of this approach are (i) explicit trans-dimensional MCMC, and (ii) superpopulation data augmentation. The superpopulation approach has the advantage of simple specification that is easily implemented in BUGS and related software. However, it reparameterizes the model so that abundance is no longer included, except as a derived quantity. This is a drawback when hierarchical models for abundance, or related parameters, are desired. Here, we analytically compare the two approaches and show that they are more closely related than might appear superficially. We exploit this relationship to specify the model in a way that allows us to include abundance as a parameter and that facilitates hierarchical modeling using readily available software such as BUGS. We use this approach to model trends in grizzly bear abundance in Yellowstone National Park from 1986 to 1998.