Yajuan Si, Natesh Pillai, and I write:
It has historically been a challenge to perform Bayesian inference in a design-based survey context. The present paper develops a Bayesian model for sampling inference using inverse-probability weights. We use a hierarchical approach in which we model the distribution of the weights of the nonsampled units in the population and simultaneously include them as predictors in a nonparametric Gaussian process regression. We use simulation studies to evaluate the performance of our procedure and compare it to the classical design-based estimator. We apply our method to the Fragile Family Child Wellbeing Study. Our studies find the Bayesian nonparametric finite population estimator to be more robust than the classical design-based estimator without loss in efficiency.
More work needs to be done for this to be a general practical tool—in particular, in the setup of this paper you only have survey weights and no direct poststratification variables—but at the theoretical level I think it’s a useful start, because it demonstrates how we can feed survey weights into a general Mister P framework in which the poststratification population sizes are unknown and need to be estimated from data. I’m very excited about this general line of research. (And we’re fitting the model in Stan.)