  The mean field variational Bayes (VB) algorithm implemented in Stan is
relatively fast and efficient, making it feasible to produce model-estimated
official statistics on a rapid timeline. Yet, while consistent point estimates
of parameters are achieved for continuous data models, the mean field
approximation often produces inaccurate uncertainty quantification to the
extent that parameters are correlated a posteriori. In this paper, we propose a
simulation procedure that calibrates uncertainty intervals for model parameters
estimated under approximate algorithms to achieve nominal coverages. Our
procedure detects and corrects biased estimation of both first and second
moments of approximate marginal posterior distributions induced by any
estimation algorithm that produces consistent first moments under specification
of the correct model. The method generates replicate datasets using parameters
estimated in an initial model run. The model is subsequently re-estimated on
each replicate dataset, and we use the empirical distribution over the
re-samples to formulate calibrated confidence intervals of parameter estimates
of the initial model run that are guaranteed to asymptotically achieve nominal
coverage. We demonstrate the performance of our procedure in Monte Carlo
simulation study and apply it to real data from the Current Employment
Statistics survey.
