Bayesian Two Stage Design Under Model Uncertainty
Traditional single stage design optimality procedures can be used to efficiently generate data for an assumed model y = f(x(m),b) + ε. The model assumptions include the form of f, the set of regressors, x(m) , and the distribution of ε. The nature of the response, y, often provides information about the model form (f) and the error distribution. It is more difficult to know, apriori, the specific set of regressors which will best explain the relationship between the response and a set of design (control) variables x. Misspecification of x(m) will result in a design which is efficient, but for the wrong model.
A Bayesian two stage design approach makes it possible to efficiently design experiments when initial knowledge of x(m) is poor. This is accomplished by using a Bayesian optimality criterion in the first stage which is robust to model uncertainty. Bayesian analysis of first stage data reduces uncertainty associated with x(m), enabling the remaining design points (second stage design) to be chosen with greater efficiency. The second stage design is then generated from an optimality procedure which incorporates the improved model knowledge. Using this approach, numerous two stage design procedures have been developed for the normal linear model. Extending this concept, a Bayesian design augmentation procedure has been developed for the purpose of efficiently obtaining data for variance modeling, when initial knowledge of the variance model is poor.