A Pólya Tree modelling framework for batch-mark data

Wildlife population surveys typically consist of multiple sampling occasions, where individuals are followed over time, enabling estimation of population size and, in open populations, of entry and exit patterns. Batch-mark (BM) surveys, where newly sampled individuals are given the same marking, often unique for each sampling occasion or each sampling period but not for each individual, provide the only viable monitoring tool for many species of amphibians, birds and fish. Modelling BM data for open populations has proven more challenging than modelling data where individuals are uniquely marked, and approaches proposed in the literature thus far rely on approximate inference or do not scale well with the number of individuals, and do not readily extend to the joint modelling of different observation processes often employed in practice. In this paper we propose a novel approach for modelling BM data, by defining a bivariate grid for modelling the latent entry and exit patterns, as well as population size. We employ the Bayesian nonparametric Pólya Tree (PT) prior for defining a model on the grid cells, which enables exact and highly efficient Bayesian inference on the number of individuals in each cell and hence of the population size and the entry/exit pattern. Our approach scales with the number of sampling occasions, instead of the number of individuals, and allows us to easily write the likelihood function for BM data under different observation processes. We demonstrate our new PT batch mark (PTBM) approach using extensive simulations and two case studies, comparing its performance with two recently proposed approaches.