  DuMouchel and Jones (1994) proposed the Bayesian D criterion by modifying the
D-optimality approach to reduce dependence of the selected design on an assumed
model. This criterion has been applied to select various single-stratum designs
for completely randomized experiments when the number of effects is greater
than the sample size. In many industrial experiments, complete randomization is
sometimes expensive or infeasible and, hence, designs used for the experiments
often have multistratum structures. However, the original Bayesian D criterion
was developed under the framework of single-stratum structures and cannot be
applied to select multistratum designs. In this paper, we study how to extend
the Bayesian approach for more complicated experiments and develop the
generalized Bayesian D criterion, which generalizes the original Bayesian D
criterion and can be applied to select single-stratum and multistratum designs
for various experiments when the number of effects is greater than the rank of
the model matrix.
