  We consider maximin and Bayesian $D$-optimal designs for nonlinear regression
models. The maximin criterion requires the specification of a region for the
nonlinear parameters in the model, while the Bayesian optimality criterion
assumes that a prior for these parameters is available. On interval parameter
spaces, it was observed empirically by many authors that an increase of
uncertainty in the prior information (i.e., a larger range for the parameter
space in the maximin criterion or a larger variance of the prior in the
Bayesian criterion) yields a larger number of support points of the
corresponding optimal designs. In this paper, we present analytic tools which
are used to prove this phenomenon in concrete situations. The proposed
methodology can be used to explain many empirically observed results in the
literature. Moreover, it explains why maximin $D$-optimal designs are usually
supported at more points than Bayesian $D$-optimal designs.
