  The United States Bureau of Labor Statistics collects data using survey
instruments under informative sampling designs that assign probabilities of
inclusion to be correlated with the response. The bureau extensively uses
Bayesian hierarchical models and posterior sampling to impute missing items in
respondent-level data and to infer population parameters. Posterior sampling
for survey data collected based on informative designs are computationally
expensive and do not support production schedules of the bureau. Motivated by
this problem, we propose a new method to scale Bayesian computations in
informative sampling designs. Our method divides the data into smaller subsets,
performs posterior sampling in parallel for every subset, and combines the
collection of posterior samples from all the subsets through their mean in the
Wasserstein space of order 2. Theoretically, we construct conditions on a class
of sampling designs where posterior consistency of the proposed method is
achieved. Empirically, we demonstrate that our method is competitive with
traditional methods while being significantly faster in many simulations and in
the Current Employment Statistics survey conducted by the bureau.
