Estimating the size of small populations from incomplete lists via graphical models

We consider the problem of estimating the size N of a closed population from q incomplete lists. Estimation of N is based on capture-recapture type models. We use graphical models to deal with possible dependencies between lists. The current parametriza-tion involves clique probabilities which have no simple concrete meaning and are delicate to manipulate in a Bayesian context insofar as hyper-Dirichlet distributions are used as priors. Our parametrization involves marginal and conditional capture probabilities. We develop our approach with q = 3. We show that there is a one-to-one and onto correspondence between both parametrizations and that placing hyper-Dirichlet distributions on the clique parameter boils down to place independent beta distributions on the capture parameters. When N is small, the non informative Bayesian analysis encounters difficulties. The posterior distribution of N may not exist for a particular graphical model: we give a necessary and sufficient condition of existence for each. Moreover, it is highly desirable that the priors on capture are compatible across the different models. Now, due to the small size of N , fulfilling this requirement demands a particular attention. We conclude by extending our approach to q = 4 lists