U-type and factorial designs for nonparametric Bayesian regression

This paper deals with the design problem for recovering a response surface by using a nonparametric Bayesian approach. The criterion for selecting the designs is based on the asymptotic average estimation variance, and three priors for the response are specified. We found the optimal design that minimizes the criterion over the lattice designs with s q-level factors and N runs. The approach we used is similar to that in Ma et al. (J. Statist. Plann. Inference 113 (2003) 323). We also obtained alternative expressions and lower bounds for the criterion corresponding to each of the three Bayes models for the two-level U-type design by using the column balance and row distance proposed in Fang et al. (J. Complexity 19 (2003) 692). These results mat be used to construct the two-level U-type designs for the nonparametric Bayesian models.