  Bayesian optimal design of experiments is a well-established approach to
planning experiments. Briefly, a probability distribution, known as a
statistical model, for the responses is assumed which is dependent on a vector
of unknown parameters. A utility function is then specified which gives the
gain in information for estimating the true value of the parameters using the
Bayesian posterior distribution. A Bayesian optimal design is given by
maximising the expectation of the utility with respect to the joint
distribution given by the statistical model and prior distribution for the true
parameter values. The approach takes account of the experimental aim via
specification of the utility and of all assumed sources of uncertainty via the
expected utility. However, it is predicated on the specification of the
statistical model. Recently, a new type of statistical inference, known as
Gibbs (or General Bayesian) inference, has been advanced. This is
Bayesian-like, in that uncertainty on unknown quantities is represented by a
posterior distribution, but does not necessarily rely on specification of a
statistical model. Thus the resulting inference should be less sensitive to
misspecification of the statistical model. The purpose of this paper is to
propose Gibbs optimal design: a framework for optimal design of experiments for
Gibbs inference. The concept behind the framework is introduced along with a
computational approach to find Gibbs optimal designs in practice. The framework
is demonstrated on exemplars including linear models, and experiments with
count and time-to-event responses.
