  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
numerical integration. 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 models can be fitted within BUGS/JAGS (commonly
used for the Bayesian complete data likelihood approach) but with significantly
improved computational efficiency compared to the commonly used
super-population data augmentation approaches (between about 10 and 77 times
more efficient in the two examples we consider). The semi-complete likelihood
approach is flexible and applicable to a range of models, including spatially
explicit capture-recapture models. The model-fitting approach is applied to two
different datasets corresponding to the closed population model $M_h$ for
snowshoe hare data and a spatially explicit capture-recapture model applied to
gibbon data.
