Scalable Method for Bayesian Experimental Design without Integrating over Posterior Distribution

We address the computational efficiency of finding the A-optimal Bayesian experimental design, where the observation map is based on partial differential equations and thus computationally expensive to evaluate. A-optimality is a widely used and easily interpreted criterion, that seeks the optimal experimental design by minimizing the expected conditional variance. Our study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integration over the Bayesian posterior distribution. In our proposed approach, we estimate the expected conditional variance via the variance of the conditional expectation and approximate the conditional expectation using its orthogonal projection property. We derive an asymptotic error estimate for the proposed estimator of the expected conditional variance and verify it with numerical experiments. Furthermore, we extend our approach to the case where the domain of the experimental design parameters is continuous. Specifically, we propose a nonlocal approximation of the conditional expectation using an artificial neural network and apply transfer learning and data augmentation to reduce the number of evaluations of the measurement model. Through numerical experiments, we demonstrate that our method greatly reduces the number of measurement model evaluations compared with widely used importance sampling-based approaches. Code is available at https://github.com/vinh-tr-hoang/DOEviaPACE.