Convergence of Pseudo Posterior Distributions under Informative Sampling

An informative sampling design assigns probabilities of inclusion that are correlated with the response of interest and induces a dependence among sampled observations. Unadjusted model-based inference performed on data acquired under an informative sampling design can be biased concerning parameters of the population generating distribution if the sample design is not accounted for in the model. Known marginal inclusion probabilities may be used to weight the likelihood contribution of each observed unit to form a "pseudo" posterior distribution with the intent to adjust for the design. This article extends a theoretical result on the consistency of the posterior distribution, defined on an analyst-specified model space, at the true generating distribution to the sampling-weighted pseudo posterior distribution used to account for an informative sampling design. We construct conditions on known marginal and pairwise inclusion probabilities that define a class of sampling designs where consistency of the pseudo posterior is achieved, in probability. We demonstrate the result on an application concerning the Bureau of Labor Statistics Job Openings and Labor Turnover Survey.