  Social and economic studies are often implemented as complex survey designs.
For example, multistage, unequal probability sampling designs utilized by
federal statistical agencies are typically constructed to maximize the
efficiency of the target domain level estimator (e.g., indexed by geographic
area) within cost constraints for survey administration. Such designs may
induce dependence between the sampled units; for example, with employment of a
sampling step that selects geographically-indexed clusters of units. A
sampling-weighted pseudo-posterior distribution may be used to estimate the
population model on the observed sample. The dependence induced between
co-clustered units inflates the scale of the resulting pseudo-posterior
covariance matrix that has been shown to induce under coverage of the
credibility sets. By bridging results across Bayesian model mispecification and
survey sampling, we demonstrate that the scale and shape of the asymptotic
distributions are different between each of the pseudo-MLE, the
pseudo-posterior and the MLE under simple random sampling. Through insights
from survey sampling variance estimation and recent advances in computational
methods, we devise a correction applied as a simple and fast post-processing
step to MCMC draws of the pseudo-posterior distribution. This adjustment
projects the pseudo-posterior covariance matrix such that the nominal coverage
is approximately achieved. We make an application to the National Survey on
Drug Use and Health as a motivating example and we demonstrate the efficacy of
our scale and shape projection procedure on synthetic data on several common
archetypes of survey designs.
