  Bayesian optimality criteria provide a robust design strategy to parameter
misspecification. We develop an approximate design theory for Bayesian
$D$-optimality for non-linear regression models with covariates subject to
measurement errors. Both maximum likelihood and least squares estimation are
studied and explicit characterisations of the Bayesian $D$-optimal saturated
designs for the Michaelis-Menten, Emax and exponential regression models are
provided. Several data examples are considered for the case of no preference
for specific parameter values, where Bayesian $D$-optimal saturated designs are
calculated using the uniform prior and compared to several other designs,
including the corresponding locally $D$-optimal designs, which are often used
in practice.
