Gibbs Optimal Design of Experiments

Bayesian optimal design is a well-established approach to planning experiments. A distribution for the responses, that is, a statistical model, is assumed, which is dependent on unknown parameters. A utility function is then specified giving gain in information in estimating the true values of the parameters, using the Bayesian posterior distribution. A Bayesian optimal design is given by maximizing expectation of the utility with respect to the distribution implied by statistical model and prior distribution for the true parameter values. The approach accounts for the experimental aim, via specification of the utility, and of assumed sources of uncertainty. However, it is predicated on the statistical model being correct. Recently, a new type of statistical inference, known as Gibbs inference, has been proposed. This is Bayesian-like, that is, uncertainty for unknown quantities is represented by a posterior distribution, but does not necessarily require specification of a statistical model. The resulting inference is less sensitive to misspecification of the statistical model. This paper introduces Gibbs optimal design: a framework for optimal design of experiments under Gibbs inference. A computational approach to find designs in practice is outlined and the framework is demonstrated on exemplars including linear models, and experiments with count and time-to-event responses.