Mathematical Sciences: Some Bayesian Problems in Sample Survey

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This proposal will consider the application of the Bayesian methodology to some problems in finite population sampling. Over the past several years the principle investigator has helped to develop a noninformative Bayesian approach to some problems in finite population sampling. It is based on the `Polya posterior' and is appropriate when one's prior beliefs about the units are roughly exchangeable. The first major goal of this project is to extend the techniques underlying the `Polya posterior' to the area of cluster sampling. This promises to yield a flexible and unified theory for an important class of problems. Recent developments in Markov chain Monte Carlo technology mean that Bayesian models which until recently have been computationally intractable can now be studied. The second major goal of this project is to use these methods to develop Bayesian models for survey sampling when one's prior beliefs about the units are no longer exchangeable. In one class of models to be considered the units will be assumed to be partially exchangeable, while in another class the populations will be assumed to be generated by a generalized urn processes. This should yield a collection of models that extends the type of prior information that can be used in an analysis in sample survey. A standard setting in survey sampling or finite population sampling assumes each member of some population possesses an unknown amount of a characteristic of interest. The statistician is interested in estimating a quantity related to this characteristic of interest. For example the characteristic of interest could be household income, the population all households in a given city and the quantity of interest the average or median household income for the city. The statistician will then use the values of the characteristic in a sample drawn from the population to estimate this quantity. The major goal of this project is to study how prior information about the population can be used to construct sensible estimates about the quantity of interest. It will use recent developments in computer simulation to investigate various models for prior information which up until now have been impossible to study.