  We investigate Bayesian predictive inference for finite population quantities
when there are unequal probabilities of selection. Only limited information
about the sample design is available; i.e., only the first-order selection
probabilities corresponding to the sample units are known. Our methodology,
unlike that of Chambers, Dorfman and Wang (1998), can be used to make inference
for finite population quantities and provides measures of precision and
intervals. Moreover, our methodology, using Markov chain Monte Carlo methods,
avoids the necessity of using asymptotic closed form approximations, necessary
for the other approaches that have been proposed. A set of simulated examples
shows that the informative model provides improved precision over a standard
ignorable model, and corrects for the selection bias.
