A Bayes-Sard Cubature Method
A Bayes-Sard Cubature Method
复制标题
DOI：
--
复制
发表时间：
2018-04
期刊：
影响因子：
--
通讯作者：
T. Karvonen;C. Oates;Simo Särkkä
中科院分区：
文献类型：
--
作者：
T. Karvonen;C. Oates;Simo Särkkä
本文的重点是数值积分的公式作为一个推理任务。迄今为止，研究工作主要集中在贝叶斯模型的发展上，其分布输出为积分提供了不确定性量化。然而，当域是高维时，与贝叶斯培养相关的点估计可能是不准确的，并且对先验非常敏感。为了解决这些缺点，我们引入了贝叶斯-萨德模型，这是一个概率框架，它结合了贝叶斯模型的灵活性和经典模型的鲁棒性。这是通过考虑被积函数的高斯过程模型来实现的，该模型的均值是一个参数回归模型，每个回归系数都有一个不适当的平坦先验。回归模型中的特征由保证精确积分的测试函数组成，剩余的自由度给予非参数部分。建立了Bayes-Sard方法的渐近收敛性，并对理论结果进行了数值验证。特别地，我们报告了在高维金融积分的背景下，与贝叶斯模型相比，误差降低了两个数量级。
This paper focusses on the formulation of numerical integration as an inferential task. To date, research effort has largely focussed on the development of Bayesian cubature, whose distributional output provides uncertainty quantification for the integral. However, the point estimators associated to Bayesian cubature can be inaccurate and acutely sensitive to the prior when the domain is high-dimensional. To address these drawbacks we introduce Bayes-Sard cubature, a probabilistic framework that combines the flexibility of Bayesian cubature with the robustness of classical cubatures which are well-established. This is achieved by considering a Gaussian process model for the integrand whose mean is a parametric regression model, with an improper flat prior on each regression coefficient. The features in the regression model consist of test functions which are guaranteed to be exactly integrated, with remaining degrees of freedom afforded to the non-parametric part. The asymptotic convergence of the Bayes-Sard cubature method is established and the theoretical results are numerically verified. In particular, we report two orders of magnitude reduction in error compared to Bayesian cubature in the context of a high-dimensional financial integral.