Bayesian Optimal Design of Experiments For Inferring The Statistical Expectation Of A Black-Box Function
This work addresses uncertainty propagation and design under uncertainty problems in engineering and science, but it is incremental as it extends existing Bayesian optimal design methods to a specific quantity of interest.
The paper tackled the problem of designing Bayesian optimal experiments to estimate the statistical expectation of a black-box function, which is common in uncertainty propagation, by deriving a semi-analytic formula for expected information gain and validating it on synthetic functions and a steel wire manufacturing case, showing improved efficiency in data acquisition.
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.