Examining posterior propriety in the Bayesian analysis of capture-recapture models

Abstract: There lies a latent danger in utilizing some known mathematical results in ecology. Some results do not apply to the problem at hand. We identify one such trend. Based on a couple of theorems in mathematical statistics, Link (2013) cautions ecologists about the inappropriateness of using the discrete uniform prior in their analysis under certain conditions and instead recommends the routine use of the scale prior during analysis. This recommendation is been absorbed immediately and widely among ecologists. In this study, we consider the two fundamental capture-recapture models used widely in ecology, $M_0$ and $M_h$, and derive conditions for posterior propriety by examining the behavior of the right tail of the posterior distributions of animal population size $N$ in a Bayesian analysis. We demonstrate that both these likelihoods are far more efficient than the ones considered in Link (2013). We argue that no particularly prescriptive approach should be adopted by ecologists in regard to choosing priors of the fear of posterior impropriety. Instead, we recommend the efficient construction of likelihoods for the problem and data on hand, choosing priors based existing knowledge of a parameter of interest and encourage examining posterior propriety by asymptotic arguments as demonstrated in this study.