MC(MC)MC: exploring Monte Carlo integration within MCMC for mark-recapture models with individual covariates

Estimating abundance from mark-recapture data is challenging when capture probabilities vary among individuals. Initial solutions to this problem were based on fitting conditional likelihoods and estimating abundance as a derived parameter. More recently, Bayesian methods using full likelihoods have been implemented via reversible jump Markov chain Monte Carlo sampling (RJMCMC) or data augmentation (DA). The latter approach is easily implemented in available software and has been applied to fit models that allow for heterogeneity in both open and closed populations. However, both RJMCMC and DA may be inefficient when modelling large populations. We describe an alternative approach using Monte Carlo (MC) integration to approximate the posterior density within a Markov chain Monte Carlo (MCMC) sampling scheme. We show how this Monte Carlo within MCMC (MCWM) approach may be used to fit a simple, closed population model including a single individual covariate and present results from a simulation study comparing RJMCMC, DA and MCWM. We found that MCWM can provide accurate inference about population size and can be more efficient than both RJMCMC and DA. The efficiency of MCWM can also be improved by using advanced MC methods like antithetic sampling. Finally, we apply MCWM to estimate the abundance of meadow voles (Microtus pennsylvanicus) at the Patuxent Wildlife Research Center in 1982 allowing for capture probabilities to vary as a function body mass.