Successive sampling discovery problems arise in finite population sampling subject to \lq size-biased' selection mechanisms. Formal statistical analysis of discovery data under such models is technically challenging. Bayesian analyses are developed here in a superpopulation framework. We show how simulation methods provide computation of posterior distributions for superpopulation parameters and, more critically, predictive inferences for unsampled units in the finite population. Model extensions cover problems of uncertainty about finite population sizes, uncertainty about sample selection mechanisms, and other practical issues. Several analyses of published oil reserve data are used for illustration

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