The Polya posterior

The Polya posterior is an objective Bayesian approach to finite population sampling. In its simplest form it assumes little prior information is available about the population and that the sample, however chosen, is "representative". That is the unobserved or unseen units in the population are assumed to be similar to the observed or seen units in the sample. It is appropriate when a classical survey sampler would be willing to use simple random sampling as their design. A good reference for the basic theory is Ghosh and Meeden (1997) Bayesian methods in finite population sampling published by Chapman and Hall.

The Polya posterior can be modified to take into account prior information about the population. A common situation is when the population mean or median of an auxiliary variable is known exactly. Less commonly either the mean or median is known to belong in some interval of values. For such problems a constrained or restricted version of the Polya posterior will yield point and interval estimates with good frequentist properties. This is discussed in a 2008 paper in Survey Methodology by Lazar, Meeden and Nelson. Here is a pdf version This research was supported in part by NSF Grant DMS 0406169.

When using the constrained Polya posterior estimators cannot be found in closed form and must be calculated by simulation using MCMC. A R packaged called polyapost is available in CRAN and can be used to find estimates as long as the sample size does not get to big. Here is a link to a introduction to polyapost a pdf file which is included with the package. It gives more detail on the kind of problems the package can handle and was used to run the simulations in the 2008 paper mentioned just above.