  We propose a Bayesian approach to estimate finite population means for small
areas. The proposed methodology improves on the traditional sample survey
methods because, unlike the traditional methods, our proposed method borrows
strength from multiple data sources. Our approach is fundamentally different
from the existing small area Bayesian approach to the finite population
sampling, which typically assumes a hierarchical model for all units of the
finite population. We assume such model only for the units of the finite
population in which the outcome variable is observed; because for these units,
the assumed model can be checked using existing statistical tools. Modeling
unobserved units of the finite population is challenging because the assumed
model cannot be checked in the absence of data on the outcome variable. To make
reasonable modeling assumptions, we propose to form several cells for each
small area using factors that potentially influence the outcome variable of
interest. This strategy is expected to bring some degree of homogeneity within
a given cell and also among cells from different small areas that are
constructed with the same factor level combination. Instead of modeling true
probabilities for unobserved individual units, we assume that population means
of cells with the same combination of factor levels are identical across small
areas and the population mean of true probabilities for a cell is identical to
the mean of true values for the observed units in that cell. We apply our
proposed methodology to a real-life COVID-19 survey, linking information from
multiple disparate data sources to estimate vaccine-hesitancy rates
(proportions) for 50 US states and Washington, D.C. (small areas). We also
provide practical ways of model selection that can be applied to a wider class
of models under similar setting but for a diverse range of scientific problems.
