  Nonprobability (convenience) samples are increasingly sought to stabilize
estimations for one or more population variables of interest that are performed
using a randomized survey (reference) sample by increasing the effective sample
size. Estimation of a population quantity derived from a convenience sample
will typically result in bias since the distribution of variables of interest
in the convenience sample is different from the population. A recent set of
approaches estimates conditional (on sampling design predictors) inclusion
probabilities for convenience sample units by specifying reference
sample-weighted pseudo likelihoods. This paper introduces a novel approach that
derives the propensity score for the observed sample as a function of
conditional inclusion probabilities for the reference and convenience samples
as our main result. Our approach allows specification of an exact likelihood
for the observed sample. We construct a Bayesian hierarchical formulation that
simultaneously estimates sample propensity scores and both conditional and
reference sample inclusion probabilities for the convenience sample units. We
compare our exact likelihood with the pseudo likelihoods in a Monte Carlo
simulation study.
