Various aspects of Bayesian inference in selection and size biased sampling problems are presented, beginning with discussion of general problems of inference in infinite and finite populations subject to selection sampling. Estimation of the size of finite populations and inference about superpopulation distributions when sampling is apparently informative is then developed in two specific problems. The first is a simple example of truncated data analysis, and some details of simulation based Bayesian analysis are presented. The second concerns {\sl discovery} sampling in which units of a finite population are selected with probabilities proportional to some measure of size. A well-known area of application is in the discovery of oil reserves, and some recently published data from this area is analysed here. Solutions to the computational problems arising are developed using iterative simulation methods. Finally, some comments are made on extensions, including multiparameter superpopulations, semi-parametric models and problems of dealing with missing data in discovery sampling.

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