  For Bayesian D-optimal design, we define a singular prior distribution for
the model parameters as a prior distribution such that the determinant of the
Fisher information matrix has a prior geometric mean of zero for all designs.
For such a prior distribution, the Bayesian D-optimality criterion fails to
select a design. For the exponential decay model, we characterize singularity
of the prior distribution in terms of the expectations of a few elementary
transformations of the parameter. For a compartmental model and several
multi-parameter generalized linear models, we establish sufficient conditions
for singularity of a prior distribution. For the generalized linear models we
also obtain sufficient conditions for non-singularity. In the existing
literature, weakly informative prior distributions are commonly recommended as
a default choice for inference in logistic regression. Here it is shown that
some of the recommended prior distributions are singular, and hence should not
be used for Bayesian D-optimal design. Additionally, methods are developed to
derive and assess Bayesian D-efficient designs when numerical evaluation of the
objective function fails due to ill-conditioning, as often occurs for
heavy-tailed prior distributions. These numerical methods are illustrated for
logistic regression.
