  We consider the problem of integrating a small probability sample (ps) and a
non-probability sample (nps). By definition, for the nps, there are no survey
weights, but for the ps, there are survey weights. The key issue is that the
nps, although much larger than the ps, can lead to a biased estimator of a
finite population quantity but with much smaller variance. We begin with a
relatively simple problem in which the population is assumed to be homogeneous
and there are no common units in the ps and the nps. We assume that there are
covariates and responses for everyone in the two samples, and there are no
covariates available for the nonsampled units. We use the nps (ps) to construct
a prior for the ps (nps). We also introduce partial discounting to avoid a
dominance of the prior. We use Bayesian predictive inference for the finite
population mean. In our illustrative example on body mass index and our
simulation study, we compare the relative performance of alternative procedures
and demonstrate that our procedure leads to improved estimates over the ps only
estimate.
