  The ranking problem is to order a collection of units by some unobserved
parameter, based on observations from the associated distribution. This problem
arises naturally in a number of contexts, such as business, where we may want
to rank potential projects by profitability; or science, where we may want to
rank variables potentially associated with some trait by the strength of the
association. Most approaches to this problem are empirical Bayesian, where we
use the data to estimate the hyperparameters of the prior distribution, then
use that distribution to estimate the unobserved parameter values. There are a
number of different approaches to this problem, based on different loss
functions for mis-ranking units. However, little has been done on the choice of
prior distribution. Typical approaches involve choosing a conjugate prior for
convenience, and estimating the hyperparameters by MLE from the whole dataset.
In this paper, we look in more detail at the effect of choice of prior
distribution on Bayesian ranking. We focus on the use of posterior mean for
ranking, but many of our conclusions should apply to other ranking criteria,
and it is not too difficult to adapt our methods to other choices of prior
distributions.
