  One of the well-known challenges in optimal experimental design is how to
efficiently estimate the nested integrations of the expected information gain.
The Gaussian approximation and associated importance sampling have been shown
to be effective at reducing the numerical costs. However, they may fail due to
the non-negligible biases and the numerical instabilities. A new approach is
developed to compute the expected information gain, when the posterior
distribution is multimodal - a situation previously ignored by the methods
aiming at accelerating the nested numerical integrations. Specifically, the
posterior distribution is approximated using a mixture distribution constructed
by multiple runs of global search for the modes and weighted local Laplace
approximations. Under any given probability of capturing all the modes, we
provide an estimation of the number of runs of searches, which is dimension
independent. It is shown that the novel global-local multimodal approach can be
significantly more accurate and more efficient than the other existing
approaches, especially when the number of modes is large. The methods can be
applied to the designs of experiments with both calibrated and uncalibrated
observation noises.
