  The estimation of probabilities of default (PDs) for low default portfolios
by means of upper confidence bounds is a well established procedure in many
financial institutions. However, there are often discussions within the
institutions or between institutions and supervisors about which confidence
level to use for the estimation. The Bayesian estimator for the PD based on the
uninformed, uniform prior distribution is an obvious alternative that avoids
the choice of a confidence level. In this paper, we demonstrate that in the
case of independent default events the upper confidence bounds can be
represented as quantiles of a Bayesian posterior distribution based on a prior
that is slightly more conservative than the uninformed prior. We then describe
how to implement the uninformed and conservative Bayesian estimators in the
dependent one- and multi-period default data cases and compare their estimates
to the upper confidence bound estimates. The comparison leads us to suggest a
constrained version of the uninformed (neutral) Bayesian estimator as an
alternative to the upper confidence bound estimators.
