使用归一化流的变分贝叶斯最优实验设计
Variational Bayesian Optimal Experimental Design with Normalizing Flows
AI总结：
提出基于归一化流的变分贝叶斯最优实验设计方法（vOED-NFs），通过条件可逆神经网络近似后验以无似然方式估计并优化EIG下界，在固定预算下实现了更低偏差且有效捕捉非高斯与多模态特征。
摘要：
贝叶斯最优实验设计（OED）寻求能最大化模型参数中预期信息增益（EIG）的实验。使用嵌套蒙特卡洛直接估计EIG计算成本高昂，且需要显式似然。相反，变分OED（vOED）通过用变分形式近似后验分布，在无需似然评估的情况下估计EIG的下界，随后通过优化其变分参数来收紧该边界。我们引入归一化流（NFs）在vOED中表示变分分布；我们将此方法称为vOED-NFs。具体而言，我们采用由耦合层组合构建的条件可逆神经网络架构的NFs，并辅以用于数据降维的摘要网络进行增强。我们提出了该下界的蒙特卡洛估计量及梯度表达式，以实现对变分参数和设计变量的基于梯度的同步优化。vOED-NFs算法随后在两个基准问题中得到验证，并应用于由偏微分方程控制的阴极电泳沉积问题，以及一个包含蚜虫种群随机建模的隐式似然案例。研究结果表明，在固定的正向模型运行预算下，与以往方法相比，4到5个耦合层的组合能够实现更低的EIG估计偏差。生成的NFs产生与真实后验高度吻合的近似后验，能够有效捕捉非高斯和多模态特征。
英文摘要：
Bayesian optimal experimental design (OED) seeks experiments that maximize the expected information gain (EIG) in model parameters. Directly estimating the EIG using nested Monte Carlo is computationally expensive and requires an explicit likelihood. Variational OED (vOED), in contrast, estimates a lower bound of the EIG without likelihood evaluations by approximating the posterior distributions with variational forms, and then tightens the bound by optimizing its variational parameters. We introduce the use of normalizing flows (NFs) for representing variational distributions in vOED; we call this approach vOED-NFs. Specifically, we adopt NFs with a conditional invertible neural network architecture built from compositions of coupling layers, and enhanced with a summary network for data dimension reduction. We present Monte Carlo estimators to the lower bound along with gradient expressions to enable a gradient-based simultaneous optimization of the variational parameters and the design variables. The vOED-NFs algorithm is then validated in two benchmark problems, and demonstrated on a partial differential equation-governed application of cathodic electrophoretic deposition and an implicit likelihood case with stochastic modeling of aphid population. The findings suggest that a composition of 4--5 coupling layers is able to achieve lower EIG estimation bias, under a fixed budget of forward model runs, compared to previous approaches. The resulting NFs produce approximate posteriors that agree well with the true posteriors, able to capture non-Gaussian and multi-modal features effectively.
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