Fast automatic Bayesian cubature using lattice sampling
Fast automatic Bayesian cubature using lattice sampling
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DOI：
10.1007/s11222-019-09895-9
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发表时间：
2018-09
影响因子：
2.2
通讯作者：
R. Jagadeeswaran;F. J. Hickernell
中科院分区：
文献类型：
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作者：
R. Jagadeeswaran;F. J. Hickernell
自动立方近似积分到用户指定的误差容限。对于高维问题，很难自适应地改变采样模式，但可以自动确定样本大小n，给定合理的固定采样模式。我们在这里使用贝叶斯观点来采用这种方法。我们假设，被积函数是一个实例的高斯随机过程参数化的一个恒定的平均值和协方差内核定义的尺度参数倍的参数化函数，指定如何在两个不同的点在域中的被积函数的值是相关的。这些超参数通过以下三种技术之一使用被积函数值进行推断或积分：经验贝叶斯，全贝叶斯或广义交叉验证。增加样本量n，直到贝叶斯后验均值的可信区间的半宽度不大于误差容限。上面概述的过程通常需要的计算成本，其中是识别超参数所需的优化步骤的数量。我们的创新是将低差异节点与匹配的协方差核配对，以降低计算成本。这种方法被证明是明确的秩-1格序列和移位不变的内核。我们的算法是在保证自动集成库（GAIL）。
Automatic cubatures approximate integrals to user-specified error tolerances. For high-dimensional problems, it is difficult to adaptively change the sampling pattern, but one can automatically determine the sample size,n, given a reasonable, fixed sampling pattern. We take this approach here using a Bayesian perspective. We postulate that the integrand is an instance of a Gaussian stochastic process parameterized by a constant mean and a covariance kernel defined by a scale parameter times a parameterized function specifying how the integrand values at two different points in the domain are related. These hyperparameters are inferred or integrated out using integrand values via one of three techniques: empirical Bayes, full Bayes, or generalized cross-validation. The sample size,n, is increased until the half-width of the credible interval for the Bayesian posterior mean is no greater than the error tolerance. The process outlined above typically requires a computational cost of, whereis the number of optimization steps required to identify the hyperparameters. Our innovation is to pair low discrepancy nodes with matching covariance kernels to lower the computational cost to. This approach is demonstrated explicitly with rank-1 lattice sequences and shift-invariant kernels. Our algorithm is implemented in the Guaranteed Automatic Integration Library (GAIL).