Survey Design, Symmetry and Posterior Distributions

Summary It is now well known that the sample design is irrelevant in a Bayesian analysis of survey data provided the units are labelled and the sampling is non-informative. However, information about the labels is often not available for the analysis, and in this situation the design will contain information about the labelling and hence affect the inferences. There is one important exception: if the prior distribution is exchangeable and we are interested only in symmetric functions of the unit values then the design is still irrelevant even without the labels.