Capture-recapture data are often collected when abundance estimation is of interest. In the presence of unobserved individual heterogeneity, specified on a continuous scale for the capture probabilities, the likelihood is not generally available in closed form, but expressible only as an analytically intractable integral. Model-fitting algorithms to estimate abundance most notably include a numerical approximation for the likelihood or use of a Bayesian data augmentation technique considering the complete data likelihood. We consider a Bayesian hybrid approach, defining a ``semi-complete'' data likelihood, composed of the product of a complete data likelihood component for individuals seen at least once within the study and a marginal data likelihood component for the individuals not seen within the study, approximated using a numerical integration approach. This approach combines the advantages of the two different approaches, with the semi-complete likelihood component specified as a single integral (over the dimension of the individual heterogeneity component). In addition, the model can be fitted within BUGS/JAGS (commonly used for the Bayesian complete data likelihood approach) but permits greater flexibility with regard to the prior specification on abundance and improved computational efficiency (compared to the commonly used super-population data augmentation approach). The semi-complete likelihood approach is applied to the closed population model $M_h$ fitted to snowshoe hare data and a spatially explicit capture-recapture model fitted to gibbon data. The performance of the algorithms are compared to the previous Bayesian super-population approach.
Keywords: BUGS; capture-recapture; closed populations; individual heterogeneity; JAGS; spatially-explicit.