The general principles of Bayesian data analysis imply that models for survey responses should be constructed conditional on all variables that affect the probability of inclusion and nonresponse, which are also the variables used in survey weighting and clustering. However, such models can quickly become very complicated, with potentially thousands of poststratification cells. It is then a challenge to develop general families of multilevel probability models that yield reasonable Bayesian inferences. We discuss in the context of several ongoing public health and social surveys. This work is currently open-ended, and we conclude with thoughts on how research could proceed to solve these problems.
"Struggles with Survey Weighting and Regression Modeling." Statist. Sci. 22 (2) 153 - 164, May 2007. https://doi.org/10.1214/088342306000000691