SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 4, Page 1310-1337, January 2020.
This work is concerned with the convergence of Gaussian process regression. A particular focus is on hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown and are learned from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. We provide a convergence analysis that (i) holds for a given, deterministic function $f$ to be emulated, and (ii) shows that convergence of Gaussian process regression is unaffected by the additional learning of hyper-parameters from data and is guaranteed in a wide range of scenarios. As the primary motivation for the work is the use of Gaussian process regression to approximate the data likelihood in Bayesian inverse problems, we provide a bound on the error introduced in the Bayesian posterior distribution in this context.

中文翻译：

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估计超参数的高斯过程回归的收敛性及其在贝叶斯逆问题中的应用

SIAM / ASA不确定性量化期刊，第8卷，第4期，第1310-1337页，2020年1月。
这项工作与高斯过程回归的收敛性有关。特别关注的是分层高斯过程回归，其中高斯过程仿真器的均值和协方差结构中出现的超参数是先验未知的，可以从数据中学习，以及后均值和协方差。我们在经验贝叶斯框架下工作，其中使用数据计算超参数的点估计，然后在后验更新之前在标准高斯过程中使用。我们提供了一个收敛分析，（i）对于要模拟的给定确定性函数$ f $成立，并且（ii）显示高斯过程回归的收敛不受从数据中额外学习超参数的影响，并且可以保证在各种各样的场景。