An Introduction to Bayesian Inference for Finite Population Characteristics

Abstract

AbstractWith an increased interest in using Bayesian methods for the analysis of sample survey data, it is timely to provide an introduction. We start from first principles, progressing to relatively simple parametric models. With the latter one can see how the likelihood and prior information are combined to make inferences. Assuming that the values of Y in the finite population come from a normal distribution with known variance, a sample of size n, and a conjugate prior distribution, explicit expressions for the posterior mean and variance of the finite population mean and variance are presented and interpreted. In a similar way, explicit expressions are given for the case where the finite population is generated from a linear regression of Y on X through the origin. This is a model typically seen in establishment surveys where \(Y_i\) and \(X_i\) represent the survey and census values for unit i. A useful extension when there is hidden cluster structure in the data is to use a Dirichlet process rather than simple parametric models such as those described in this chapter. Multiple regression with post-stratification (MRP) is based on the use of many categorical variables and specialized hierarchical priors. MRP has seen widespread application, especially when the data are from nonprobability samples or probability samples with low response rates. Finally, there is an extensive discussion of alternative (Bayesian) inferential methods when the data are categorical. This is an attractive option when one wishes to avoid postulating parametric continuous distributions.KeywordsCategorical dataDirichlet processPost-stratificationSurvey sampling