The Minnesota Population Center (MPC) is an interdepartmental demography research group at the University of Minnesota.  A major 
goal of the MPC is to create databases of demographic information which can be utilized in the study of economic and social behavior.  
The demographic information generally is collected from national surveys, such as the U.S. Census Bureau Long Form and the American 
Community Survey.

The most common statistical problems encountered by MPC data users are population mean estimation and regression coefficient 
estimation.  In performing such estimation, these researchers seek to apply statistical methods with good Frequentist properties, but 
the methods must also be practical.  Most MPC users have Ph.D.'s in the social sciences, but they do not have advanced statistical training.

The Polya Posterior is a Bayesian mode of survey analysis which has been previously shown to have good Frequentist properties.  
However, implementation of the Polya Posterior requires knowledge of linear algebra, Markov Chain Monte Carlo, and other advanced 
programming skills.  In this paper we attempt to make the Polya Posterior more practical for MPC users by considering an extension to 
cluster sampling, and by considering various approximations to the Polya Posterior.  One approximation is based upon a new concept 
called *Bayesian sampling weights*.  Not only are such weights practical to use, but, when estimating the population mean in the presence 
of auxiliary information, the weights can have better Frequentist properties than the Polya Posterior.  Such behavior is illustrated with 
simulations conducted upon MPC data.  Theoretical support for the Bayesian sampling weights is provided through an admissibility argument.