Bayesian decision-theoretic design of experiments under an alternative model

Traditionally Bayesian decision-theoretic design of experiments proceeds by choosing a design to minimise expectation of a given loss function over the space of all designs. The loss function encapsulates the aim of the experiment, and the expectation is taken with respect to the joint distribution of all unknown quantities implied by the statistical model that will be fitted to observed responses. In this paper, an extended framework is proposed whereby the expectation of the loss is taken with respect to a joint distribution implied by an alternative statistical model. Motivation for this includes promoting robustness, ensuring computational feasibility and for allowing realistic prior specification when deriving a design. To aid in exploring the new framework, an asymptotic approximation to the expected loss under an alternative model is derived, and the properties of different loss functions are established. The framework is then demonstrated on a linear regression versus full-treatment model scenario, on estimating parameters of a non-linear model under model discrepancy and a cubic spline model under an unknown number of basis functions.