  Finding the optimal design of experiments in the Bayesian setting typically
requires estimation and optimization of the expected information gain
functional. This functional consists of one outer and one inner integral,
separated by the logarithm function applied to the inner integral. When the
mathematical model of the experiment contains uncertainty about the parameters
of interest and nuisance uncertainty, (i.e., uncertainty about parameters that
affect the model but are not themselves of interest to the experimenter), two
inner integrals must be estimated. Thus, the already considerable computational
effort required to determine good approximations of the expected information
gain is increased further. The Laplace approximation has been applied
successfully in the context of experimental design in various ways, and we
propose two novel estimators featuring the Laplace approximation to alleviate
the computational burden of both inner integrals considerably. The first
estimator applies Laplace's method followed by a Laplace approximation,
introducing a bias. The second estimator uses two Laplace approximations as
importance sampling measures for Monte Carlo approximations of the inner
integrals. Both estimators use Monte Carlo approximation for the remaining
outer integral estimation. We provide four numerical examples demonstrating the
applicability and effectiveness of our proposed estimators.
