分层贝叶斯求积法
Hierarchical Bayesian Quadrature
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中文总结 AI 辅助
研究针对数值积分中贝叶斯求积法因基于平稳协方差函数导致对非平稳被积函数模型错误指定的问题，提出通过积分域树状划分成局部平稳模型的方法，经GP条件层次结构重组估计，提升了非平稳被积函数积分效果。
中文摘要 AI 辅助
数值积分是各种科学计算应用的基石，如工程模拟和概率机器学习中的模型证据计算。贝叶斯求积法使用高斯过程代理，明确编码关于被积函数的结构假设，以获得具有量化不确定性的积分估计。这些代理主要基于平稳协方差函数，导致对表现出非平稳行为的被积函数出现模型错误指定。我们通过将积分域自适应增长、基于树的划分为局部平稳模型来解决这个问题。我们的方法通过GP条件层次结构重新组合局部积分估计，重新引入跨子域相关性，同时模型选择标准控制树的增长以避免不必要的划分。所得算法简单，无需MCMC，并根据局部被积函数复杂性调整其评估预算。在基准积分问题和一个流行病学模型的模型证据计算中，分层贝叶斯求积法在非平稳被积函数上比标准贝叶斯求积法有显著提升，同时在平稳被积函数上性能相当。
英文摘要
Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.
发表机构
- Tübingen AI Center（图宾根人工智能中心）
- University of Tübingen（图宾根大学）
- School of Engineering Sciences（工程科学学院）
- Lappeenranta–Lahti University of Technology（拉佩伦塔-拉赫蒂技术大学）
机构由 AI 辅助整理，请以论文原文为准。
