  Bayesian estimation is increasingly popular for performing model based
inference to support policymaking. These data are often collected from surveys
under informative sampling designs where subject inclusion probabilities are
designed to be correlated with the response variable of interest. Sampling
weights constructed from marginal inclusion probabilities are typically used to
form an exponentiated pseudo likelihood that adjusts the population likelihood
for estimation on the sample due to ease-of-estimation. We propose an
alternative adjustment based on a Bayes rule construction that simultaneously
performs weight smoothing and estimates the population model parameters in a
fully Bayesian construction. We formulate conditions on known marginal and
pairwise inclusion probabilities that define a class of sampling designs where
$L_{1}$ consistency of the joint posterior is guaranteed. We compare
performances between the two approaches on synthetic data, which reveals that
our fully Bayesian approach better estimates posterior uncertainty without a
requirement to calibrate the normalization of the sampling weights. We
demonstrate our method on an application concerning the National Health and
Nutrition Examination Survey exploring the relationship between caffeine
consumption and systolic blood pressure.
