论文标题
通用贝叶斯的方法解决模型错误的反问题
Generalized Bayes Approach to Inverse Problems with Model Misspecification
论文作者
论文摘要
我们提出了一个通用框架，用于获得基于PDE的反问题的概率解决方案。贝叶斯方法对于不确定性定量具有吸引力，但假设可能了解可能性模型或数据生成过程。在许多反问题中，很难证明此假设是合理的，因为数据生成过程的规范并不明显。我们采用吉布斯后框架，该框架直接在参数的概率分布空间上提出了正则化问题。我们提出了一个新颖的模型比较框架，该框架根据其“预测性能”评估给定损失的最佳性。我们提供交叉验证程序来校准变分目标的正则化参数并比较多个损失函数。还提出了一些新型的吉布斯后期理论特性。我们通过模拟示例说明了我们框架的实用性，该示例是由用于表征超声振动学中动脉血管的基于分散的波模型的动机。
We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its "predictive performance". We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.
