  Bayesian optimal design of experiments (BODE) has been successful in
acquiring information about a quantity of interest (QoI) which depends on a
black-box function. BODE is characterized by sequentially querying the function
at specific designs selected by an infill-sampling criterion. However, most
current BODE methods operate in specific contexts like optimization, or
learning a universal representation of the black-box function. The objective of
this paper is to design a BODE for estimating the statistical expectation of a
physical response surface. This QoI is omnipresent in uncertainty propagation
and design under uncertainty problems. Our hypothesis is that an optimal BODE
should be maximizing the expected information gain in the QoI. We represent the
information gain from a hypothetical experiment as the Kullback-Liebler (KL)
divergence between the prior and the posterior probability distributions of the
QoI. The prior distribution of the QoI is conditioned on the observed data and
the posterior distribution of the QoI is conditioned on the observed data and a
hypothetical experiment. The main contribution of this paper is the derivation
of a semi-analytic mathematical formula for the expected information gain about
the statistical expectation of a physical response. The developed BODE is
validated on synthetic functions with varying number of input-dimensions. We
demonstrate the performance of the methodology on a steel wire manufacturing
problem.
