Estimating posteriors and the associated model evidences, with desired accuracy and affordable computational cost, is a core issue of Bayesian model updating, and can be of great challenge given expensive-to-evaluate models and posteriors with complex features such as multi-modalities of unequal importance, nonlinear dependencies and high sharpness. Bayesian Quadrature (BQ) equipped with active learning has emerged as a competitive framework for tackling this challenge, as it provides flexible balance between computational cost and accuracy. The performance of a BQ scheme is fundamentally dictated by the acquisition function as it exclusively governs the active generation of integration points. After reexamining one of the most advanced acquisition function from a prospective inference perspective and reformulating the quadrature rules for prediction, four new acquisition functions, inspired by distinct intuitions on expected rewards, are primarily developed, all of which are accompanied by elegant interpretations and highly efficient numerical estimators. Mathematically, these four acquisition functions measure, respectively, the prediction uncertainty of posterior, the contribution to prediction uncertainty of evidence, as well as the expected reduction of prediction uncertainties concerning posterior and evidence, and thus provide flexibility for highly effective design of integration points. These acquisition functions are further extended to the transitional BQ scheme, along with several specific refinements, to tackle the above-mentioned challenges with high efficiency and robustness. Effectiveness of the developments is ultimately demonstrated with extensive benchmark studies and application to an engineering example.

翻译：以期望的精度和可承受的计算成本估计后验分布及相关的模型证据，是贝叶斯模型更新的核心问题。在模型评估代价高昂且后验分布具有复杂特征（如重要性不等的多峰性、非线性依赖关系和高锐度）的情况下，这一问题可能极具挑战性。配备主动学习的贝叶斯求积法已成为应对这一挑战的竞争性框架，因其在计算成本与精度之间提供了灵活的平衡。BQ方案的性能从根本上由获取函数决定，因为它专门控制着积分点的主动生成。在从前瞻性推断的角度重新审视最先进的获取函数之一并重新表述用于预测的求积规则后，主要发展了四种新的获取函数，它们均受到关于期望收益的不同直觉启发，并都配有优雅的解释和高效的数值估计器。从数学上讲，这四种获取函数分别度量后验的预测不确定性、证据对预测不确定性的贡献，以及关于后验和证据的预测不确定性的期望减少量，从而为积分点的高效设计提供了灵活性。这些获取函数进一步扩展到过渡性BQ方案中，并结合若干具体改进，以高效且稳健地应对上述挑战。最终，通过广泛的基准研究和一个工程实例的应用，证明了所提方法的有效性。