Optimal experimental design (OED) provides a systematic approach to quantify and maximize the value of experimental data. Under a Bayesian approach, conventional OED maximizes the expected information gain (EIG) on model parameters. However, we are often interested in not the parameters themselves, but predictive quantities of interest (QoIs) that depend on the parameters in a nonlinear manner. We present a computational framework of predictive goal-oriented OED (GO-OED) suitable for nonlinear observation and prediction models, which seeks the experimental design providing the greatest EIG on the QoIs. In particular, we propose a nested Monte Carlo estimator for the QoI EIG, featuring Markov chain Monte Carlo for posterior sampling and kernel density estimation for evaluating the posterior-predictive density and its Kullback-Leibler divergence from the prior-predictive. The GO-OED design is then found by maximizing the EIG over the design space using Bayesian optimization. We demonstrate the effectiveness of the overall nonlinear GO-OED method, and illustrate its differences versus conventional non-GO-OED, through various test problems and an application of sensor placement for source inversion in a convection-diffusion field.

翻译：最优实验设计（OED）提供了一种系统性方法，用于量化并最大化实验数据的价值。在贝叶斯框架下，传统OED方法以最大化模型参数的期望信息增益（EIG）为目标。然而，我们关注的往往并非参数本身，而是依赖于参数的非线性预测量（QoIs）。针对非线性观测与预测模型，本文提出了一种预测型目标导向OED（GO-OED）计算框架，该框架能够找到使QoIs的EIG最大化实验设计方案。具体而言，我们提出了一种适用于QoI-EIG的嵌套蒙特卡罗估计量，该估计量采用马尔可夫链蒙特卡罗方法进行后验采样，并通过核密度估计评估后验预测密度及其与先验预测之间的KL散度。进而，利用贝叶斯优化方法在实验设计空间中最大化EIG，从而确定GO-OED方案。通过多个测试问题以及对流-扩散场中源项反演的传感器布设应用，我们验证了所提出的非线性GO-OED方法的有效性，并揭示了其与传统非GO-OED方法之间的差异性。