Bayesian (probabilistic) model updating is a fundamental concept in computational science, allowing for the incorporation of prior beliefs with observed data to reduce prediction uncertainty of a computer simulator. However, the efficient evaluation of posterior probability density functions (PDFs) of model parameters poses challenges, particularly for computationally expansive simulators. This work presents a sampling-based adaptive Bayesian quadrature method to fill this gap. The method is based on approximating the simulator under investigation with a Gaussian process (GP) model, and then a conditional sampling procedure is introduced for generating sample paths, this way to infer a probability distribution for the evidence term. This inferred probability distribution indeed measures the prediction uncertainty of the evidence term, and thus based on which, an acquisition function is proposed to identify the site at which the prediction uncertainty of the GP model contributes the most to that of the evidence term. All the above ingredients finally form an adaptive algorithm for updating the posterior PDFs of model parameters with pre-specified accuracy tolerance. Case studies across numerical examples and engineering applications validate the ability of the proposed method to deal with multi-modal problems, and demonstrate its superiority in terms of computational efficiency and precision for estimating model evidence and posterior PDFs.

中文翻译：

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基于采样的自适应贝叶斯求积，用于概率模型更新


贝叶斯（概率）模型更新是计算科学中的一个基本概念，它允许将先验信念与观察到的数据相结合，以减少计算机模拟器的预测不确定性。然而，对模型参数的后验概率密度函数 （PDF） 的有效评估带来了挑战，特别是对于计算扩展的模拟器。这项工作提出了一种基于采样的自适应贝叶斯正交方法来填补这一空白。该方法基于使用高斯过程 （GP） 模型对正在调查的模拟器进行近似，然后引入条件抽样程序来生成样本路径，从而推断证据项的概率分布。这种推断的概率分布确实衡量了证据项的预测不确定性，因此，在此基础上，提出了一个采集函数来识别 GP 模型的预测不确定性对证据项的预测不确定性贡献最大的位点。以上所有成分最终形成了一个自适应算法，用于以预先指定的精度公差更新模型参数的后验 PDF。数值示例和工程应用的案例研究验证了所提出的方法处理多模态问题的能力，并证明了它在估计模型证据和后验 PDF 的计算效率和精度方面的优越性。