  In population studies, it is standard to sample data via designs in which the
population is divided into strata, with the different strata assigned different
probabilities of inclusion. Although there have been some proposals for
including sample survey weights into Bayesian analyses, existing methods
require complex models or ignore the stratified design underlying the survey
weights. We propose a simple approach based on modeling the distribution of the
selected sample as a mixture, with the mixture weights appropriately adjusted,
while accounting for uncertainty in the adjustment. We focus for simplicity on
Dirichlet process mixtures but the proposed approach can be applied more
broadly. We sketch a simple Markov chain Monte Carlo algorithm for computation,
and assess the approach via simulations and an application.
