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Bayesian optimal design of experiments (BODE) has been successful inacquiring information about a quantity of interest (QoI) which depends on ablack-box function. BODE is characterized by sequentially querying the functionat specific designs selected by an infill-sampling criterion. However, mostcurrent BODE methods operate in specific contexts like optimization, orlearning a universal representation of the black-box function. The objective ofthis paper is to design a BODE for estimating the statistical expectation of aphysical response surface. This QoI is omnipresent in uncertainty propagationand design under uncertainty problems. Our hypothesis is that an optimal BODEshould be maximizing the expected information gain in the QoI. We represent theinformation gain from a hypothetical experiment as the Kullback-Liebler (KL)divergence between the prior and the posterior probability distributions of theQoI. The prior distribution of the QoI is conditioned on the observed data andthe posterior distribution of the QoI is conditioned on the observed data and ahypothetical experiment. The main contribution of this paper is the derivationof a semi-analytic mathematical formula for the expected information gain aboutthe statistical expectation of a physical response. The developed BODE isvalidated on synthetic functions with varying number of input-dimensions. Wedemonstrate the performance of the methodology on a steel wire manufacturingproblem.

Bayesian Optimal Design of Experiments For Inferring The Statistical Expectation Of A Black-Box Function