  A Bayesian design is given by maximising an expected utility over a design
space. The utility is chosen to represent the aim of the experiment and its
expectation is taken with respect to all unknowns: responses, parameters and/or
models. Although straightforward in principle, there are several challenges to
finding Bayesian designs in practice. Firstly, the utility and expected utility
are rarely available in closed form and require approximation. Secondly, the
design space can be of high-dimensionality. In the case of intractable
likelihood models, these problems are compounded by the fact that the
likelihood function, whose evaluation is required to approximate the expected
utility, is not available in closed form. A strategy is proposed to find
Bayesian designs for intractable likelihood models. It relies on the
development of an automatic, auxiliary modelling approach, using multivariate
Gaussian process emulators, to approximate the likelihood function. This is
then combined with a copula-based approach to approximate the marginal
likelihood (a quantity commonly required to evaluate many utility functions).
These approximations are demonstrated on examples of stochastic process models
involving experimental aims of both parameter estimation and model comparison.
