4 Constrained Polya posterior weights

Jeremy Strief and Glen Meeden

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A possible criticism of the Polya posterior and the CPP is that any simulated full copy of the population will only contain values of the characteristic that appeared in the sample. But it is exactly this property that will allow us to attach weights to the members of the sample.

We assume that we have a fixed sample for which the subset of the simplex defined by equations (3.1) and (3.2) is nonempty. For $j=1,\dots ,n$ let

$w_j=NE\left(p_j\right)=N{\mu }_j$ (4.1)

where the expectation is taken with respect to the CPP. Note that the sum of the elements of $w=\left(w_1,\dots ,w_n\right)$ is the population size $N$ and $w_j$ can be thought of as the weight associated with the $j^{\text{th}}$member of the sample. These weights depend only on the observed values of the auxiliary variables and the known population constraints. Hence this is a stepwise Bayes method of attaching weights to the units in the sample which incorporates the prior information present in the auxiliary variables and does not depend explicitly on the sampling design.

We are assuming here that the population size $N$ is know which may not always be the case. In such situations one could replace $N$in the above equation by an estimate. If the estimate is a good one then the resulting inferences for a population total should be satisfactory. When estimating a population mean the results would be much less sensitive to how close the estimate is to the true population size.

Much survey data which are used by social science researchers comes with weights attached to individual units. In such cases the CPP weights could be attached in the same way and the user would not need to use MCMC methods to calculate the weights. We will use the weights to define the Weighted Dirichlet posterior that can be used to find point and interval estimates of population quantities of interest at a relative modest computational cost. In the rest of the paper we will give examples to show that these weights can be used to generate inferential procedures with good frequentist properties.

But before proceeding we make a simple observation. Suppose we have in hand the sample along with a set of weights. If $N$ is large, then we can construct a population where the proportion of units in the population of type $\left(y_i,x_i\right)$ is $w_i/N$ for $i=1,\dots ,n.$ Given the sample and the set of weights, we can think of this constructed population as the best guess for the unknown population. Then

${\overline{y}}_{bw}={\displaystyle \sum _{i=1}^n\frac{w_i}{N}{y}_i}$ and ${\sigma }_{bw}^2={\displaystyle \sum _{i=1}^n\frac{w_i}{N}}{\left(y_i-{\overline{y}}_{bw}\right)}^2$ (4.2)

are the mean and variance of this constructed population.

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