论文标题
高斯工艺插值中平滑性参数估计的渐近界限
Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation
论文作者
论文摘要
通常，将确定性响应函数建模，例如计算机实验的输出，作为具有Matérn协方差内核的高斯过程。 Matérn内核的平滑度参数在较大的数据限制下确定了模型的许多重要特性，包括条件平均值对响应函数的收敛速率。我们证明，当数据是在$ \ mathbb {r}^d $的固定有限子集上获得的，平滑度参数的最大似然估计不能渐近地降低真相。也就是说，如果数据生成的响应功能具有Sobolev平滑度$ν_0> d/2 $，则平滑度参数估算不能渐近地小于$ν_0$。下边界很锋利。此外，我们表明，最大似然估计恢复了一类紧凑的自相似功能的真实平滑度。对于交叉验证，我们证明了一个渐近下限$ν_0 -d/2 $，但是不太可能是锋利的。结果基于Sobolev空间中的近似理论和一些限制参数估计器所需的值集的一般定理。
It is common to model a deterministic response function, such as the output of a computer experiment, as a Gaussian process with a Matérn covariance kernel. The smoothness parameter of a Matérn kernel determines many important properties of the model in the large data limit, including the rate of convergence of the conditional mean to the response function. We prove that the maximum likelihood estimate of the smoothness parameter cannot asymptotically undersmooth the truth when the data are obtained on a fixed bounded subset of $\mathbb{R}^d$. That is, if the data-generating response function has Sobolev smoothness $ν_0 > d/2$, then the smoothness parameter estimate cannot be asymptotically less than $ν_0$. The lower bound is sharp. Additionally, we show that maximum likelihood estimation recovers the true smoothness for a class of compactly supported self-similar functions. For cross-validation we prove an asymptotic lower bound $ν_0 - d/2$, which however is unlikely to be sharp. The results are based on approximation theory in Sobolev spaces and some general theorems that restrict the set of values that the parameter estimators can take.
