2 The Polya posterior

Jeremy Strief and Glen Meeden

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Let $s$ be the set of labels of a sample of size $n$ from a population of size $N.$ For convenience we assume the members of $s$ are $\mathrm{1,2,}\dots \mathrm{,}n$ and we also suppose that $n/N$ is very small. Let $y=\left(y_1,y_2,\dots ,y_N\right)$ be the characteristic of interest and $y_s$ be the observed sample values.

The Polya posterior is based upon Polya sampling from an urn. Polya sampling works as follows: suppose that the values from $n$ observed or seen units are marked on $n$ balls and placed in urn 1. The remaining unseen $N-n$ units of the population are represented by $N-n$ unmarked balls placed in urn 2. One ball from each urn is drawn with equal probability, and the ball from urn 2 is assigned the value of the ball from urn 1. Both balls are then returned to urn 1. Thus at the second stage of Polya sampling, urn 1 has $n+1$ balls and urn 2 has $N-n-1$ balls. This procedure is repeated until urn 2 is empty, at which point the $N$ balls in urn 1 constitute one complete simulated copy of the population. Any finite population quantity $–$ means, totals, quantiles, regression coefficients $–$ may now be calculated from the complete copy. For the population quantity of interest we may simulate $K$such complete copies and in each case calculate its value. The mean of these simulated values is the point estimate and an approximate 95% Bayesian credible interval is given by the 2.5% and 97.5% quantiles of the values.

One can check that under the Polya posterior the posterior expectation of the population mean is just the sample mean and the posterior variance is just $\left(n-1\right)/\left(n+1\right)$times the usual design based variance of the sample mean under simple random sampling without replacement. The Polya posterior has a decision theoretic justification based on its stepwise Bayes nature. Using this fact many standard estimators can be shown to be admissible. Details can be found in Ghosh and Meeden (1997). The Polya posterior is the Bayesian bootstrap of Rubin (1981) applied to finite population sampling. Lo (1988) also discusses the Bayesian bootstrap in finite population sampling. Some early related work can be found in Hartley and Rao (1968) and Binder (1982).

For the sample unit $i$ let $p_i$ denote the proportion of units in a full, simulated copy of the population which have the value $y_i.$ Ghosh and Meeden (1997) showed that under the Polya posterior $E\left(p_i\right)=1/n.$ If we let

$w_i=NE\left(p_i\right)=N/n$

then $w_i$ can be interpreted as the weight attached to unit $i$ since it equals the average number of units in the population represented by unit $i,$ under the Polya posterior. Recall that under simple random sampling without replacement $n/N$is the inclusion probability for each unit. Hence in this case the usual frequentist weight, which is the reciprocal of the inclusion probability, and Polya posterior weight defined above agree.

So in situations of limited prior information the Polya posterior yields weights identical to frequentist weights derived from the design of simple random sampling without replacement. The Polya posterior justification for these weights does not depend explicitly on the design and would be appropriate anytime the sampler believes the observed and unobserved units in the population are roughly exchangeable.

We next address the issue of the relationship of the Polya posterior with usual bootstrap methods in finite population sampling. Both approaches are based on an assumption of exchangeability. Gross (1980) introduced the basic idea for the bootstrap. Assume simple random sampling without replacement and suppose it is the case that $N/n=m$ is an integer. Given a sample we create a good guess for the population by combining $m$ replicates of the sample. By taking repeated random samples of size $n$ from this created population we can study the behavior of an estimator of interest. Booth, Bulter and Hall (1994) studied the asymptotic properties of such estimators. Hu, Zhang, Cohen and Salvucci (1997) is an example where the sample was used to construct an artificial population and then repeated samples were drawn from the constructed population to construct an estimate of the variance of their estimator and to construct confidence intervals.

Note this is in contrast to the Polya posterior which considers the sample fixed and repeatedly generates complete versions of the population.

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