Bayesian estimation under informative sampling

Bayesian analysis is increasingly popular for use in social science and other application areas where the data are observations from an informative sample. An informative sampling design leads to inclusion probabilities that are correlated with the response variable of interest. Model inference performed on the observed sample taken from the population will be biased for the population generative model under informative sampling since the balance of information in the sample data is different from that for the population. Typical approaches to account for an informative sampling design under Bayesian estimation are often difficult to implement because they require re-parameterization of the hypothesized generating model, or focus on design, rather than model-based, inference. We propose to construct a pseudo-posterior distribution that utilizes sampling weights based on the marginal inclusion probabilities to exponentiate the likelihood contribution of each sampled unit, which weights the information in the sample back to the population. Our approach provides a nearly automated estimation procedure applicable to any model specified by the data analyst for the population and retains the population model parameterization and posterior sampling geometry. We construct conditions on known marginal and pairwise inclusion probabilities that define a class of sampling designs where $L_{1}$ consistency of the pseudo posterior is guaranteed. We demonstrate our method on an application concerning the Bureau of Labor Statistics Job Openings and Labor Turnover Survey.