Michael Axelrod writes:

Do you have any recommendations for articles and books on survey sampling using Bayesian methods?

The whole subject of survey sampling seems not quite in the mainstream of statistics. They have model-based and designed-based sampling strategies, which give rise to 4 combinations. Do Bayesian methods impact both strategies?

My quick answer is that you can fit your usual Bayesian regression models. Just make sure to condition on all variables that affect the probability of inclusion in the sample. Of course you won’t really know what these variables are, but a quick start is to use whatever variables are used in the survey weights (if these are provided). You might be adjusting for a lot of variables, so you might want to fit a multilevel regression–that’s usually the point of doing Bayes in the first place. And then you have to average up your estimates to get inferences about the population. That’s poststratification. Put it together and you have multilevel regression and poststratification: Mister P.

To answer your original question of what to read on this: No books, really–well, maybe my two books. They’re strong on Bayes but don’t really focus on survey methods. We do have some survey-analysis examples, though.

For something on the theoretical side, there’s this article. For something more methods-y, this article by Lax and Phillips. Or this article that shows Mister P in application.

Perhaps commenters have other suggested readings.