  The construction of decision-theoretic Bayesian designs for
realistically-complex nonlinear models is computationally challenging, as it
requires the optimization of analytically intractable expected utility
functions over high-dimensional design spaces. We provide the most general
solution to date for this problem through a novel approximate coordinate
exchange algorithm. This methodology uses a Gaussian process emulator to
approximate the expected utility as a function of a single design coordinate in
a series of conditional optimization steps. It has flexibility to address
problems for any choice of utility function and for a wide range of statistical
models with different numbers of variables, numbers of runs and randomization
restrictions. In contrast to existing approaches to Bayesian design, the method
can find multi-variable designs in large numbers of runs without resorting to
asymptotic approximations to the posterior distribution or expected utility.
The methodology is demonstrated on a variety of challenging examples of
practical importance, including design for pharmacokinetic models and design
for mixed models with discrete data. For many of these models, Bayesian designs
are not currently available. Comparisons are made to results from the
literature, and to designs obtained from asymptotic approximations.
