2. Generating synthetic populations from survey data

Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan

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The basic concept of Bayesian finite population inference involves imputing the non-sampled values of the population from the posterior predictive distribution based on the observed data. Assume the population values are $Y=\left(Y_1,\dots ,Y_N\right)$ and the observed data, $Y_{\text{obs}}=\left(y_1,\dots ,y_n\right)$ is obtained in a survey with sampling indicators $I=\left(I_1,\dots ,I_N\right).$ The Bayesian population inference allows for the use of parametric model $\mathrm{Pr}\left(Y|\theta \right)$ for population data based on the posterior predictive distribution for the unobserved elements of the population $\mathrm{Pr}\left(Y_{\text{nob}}|Y_{\text{obs}}\right):$

$$\mathrm{Pr}\left(Y_{\text{nob}}|Y_{\text{obs}}\right)={{\displaystyle \int }}^{\text{}}\text{Pr}\left(Y_{\text{nob}}|Y_{\text{obs}},\theta \right)\mathrm{Pr}\left(\theta |Y_{\text{obs}}\right)d\theta$$

(Ericson 1969; Little 1993; Rubin 1987; Scott 1977; Skinner et al. 1989). Here we use the model $\mathrm{Pr}\left(Y|\theta \right)$ to approximate the entire population distribution $\mathrm{Pr}\left(Y\right)$ and average over the posterior distribution based on the sampled data $\mathrm{Pr}\left(\text{θ}|Y_{\text{obs}}\right).$ In the case that there are design variables known for the entire population available, the above model can be naturally extended by conditioning on these variables.

Implicit in the derivation of above is that the sampling indicator $I$ need not be modeled. This requires ignorable sampling (Rubin 1987) (the distribution of $I$ does not depend on unobserved data), as well as a model for the data $\mathrm{Pr}\left(Y|\theta \right)$ that is attentive to design features and robust enough to sufficiently capture all relevant aspects of the distribution of $Y$ of interest. Our goal here is to develop a method to generate draws from $\mathrm{Pr}\left(Y_{\text{nob}}|Y_{\text{obs}}\right)$ that account for all the design features in $Y_{\text{obs}}$ so that draws from the posterior distribution of $Y_{\text{nob}}|Y_{\text{obs}}$ can be treated as a simple random sample in analysis.

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