  This paper explores Bayesian inference for a biased sampling model in
situations where the population of interest cannot be sampled directly, but
rather through an indirect and inherently biased method. Observations are
viewed as being the result of a multinomial sampling process from a tagged
population which is, in turn, a biased sample from the original population of
interest. This paper presents several Gibbs Sampling techniques to estimate the
joint posterior distribution of the original population based on the observed
counts of the tagged population. These algorithms efficiently sample from the
joint posterior distribution of a very large multinomial parameter vector.
Samples from this method can be used to generate both joint and marginal
posterior inferences. We also present an iterative optimization procedure based
upon the conditional distributions of the Gibbs Sampler which directly computes
the mode of the posterior distribution. To illustrate our approach, we apply it
to a tagged population of messanger RNAs (mRNA) generated using a common
high-throughput technique, Serial Analysis of Gene Expression (SAGE).
Inferences for the mRNA expression levels in the yeast Saccharomyces cerevisiae
are reported.
