  Complex models used to describe biological processes in epidemiology and
ecology often have computationally intractable or expensive likelihoods. This
poses significant challenges in terms of Bayesian inference but more
significantly in the design of experiments. Bayesian designs are found by
maximising the expectation of a utility function over a design space, and
typically this requires sampling from or approximating a large number of
posterior distributions. This renders approaches adopted in inference
computationally infeasible to implement in design. Consequently, optimal design
in such fields has been limited to a small number of dimensions or a restricted
range of utility functions. To overcome such limitations, we propose a
synthetic likelihood-based Laplace approximation for approximating utility
functions for models with intractable likelihoods. As will be seen, the
proposed approximation is flexible in that a wide range of utility functions
can be considered, and remains computationally efficient in high dimensions. To
explore the validity of this approximation, an illustrative example from
epidemiology is considered. Then, our approach is used to design experiments
with a relatively large number of observations in two motivating applications
from epidemiology and ecology.
