A predictive approach to the Bayesian design problem with application to normal regression models

A predictive decision-theoretic approach is developed for the Bayesian design problem. The loss functions used are fair Bayes, or proper scoring rules, and are quadratic measures of distance between probability measures. Optimal Bayesian designs are those which minimise the preposterior risk for the decision problem. Such designs typically depend on both the prior distribution and the loss function. The results are applied to certain normal regression models where explicit optimal designs are constructed.