A common approach to approximating Gaussian log-likelihoods at scale exploits the fact that precision matrices can be well-approximated by sparse matrices in some circumstances. This strategy is motivated by the \emph{screening effect}, which refers to the phenomenon in which the linear prediction of a process $Z$ at a point $\mathbf{x}_0$ depends primarily on measurements nearest to $\mathbf{x}_0$. But simple perturbations, such as i.i.d. measurement noise, can significantly reduce the degree to which this exploitable phenomenon occurs. While strategies to cope with this issue already exist and are certainly improvements over ignoring the problem, in this work we present a new one based on the EM algorithm that offers several advantages. While in this work we focus on the application to Vecchia's approximation (1988), a particularly popular and powerful framework in which we can demonstrate true second-order optimization of M steps, the method can also be applied using entirely matrix-vector products, making it applicable to a very wide class of precision matrix-based approximation methods.

翻译：在大规模近似高斯对数似然时，一种常见策略是利用精度矩阵在某些情况下可被稀疏矩阵良好逼近的特性。该策略基于"筛选效应"，即过程$Z$在点$\mathbf{x}_0$处的线性预测主要依赖于距离$\mathbf{x}_0$最近的观测值。然而，简单的扰动（如独立同分布的测量噪声）会显著降低这一可被利用现象的发生程度。尽管已有解决该问题的策略，且这些方法无疑优于忽视该问题的做法，但本文提出了一种基于EM算法的新方法，具备多项优势。虽然本文聚焦于Vecchia近似（1988）这一特别流行的强大框架，并在该框架中展示了M步的真正的二阶优化，但该方法亦可完全通过矩阵-向量乘积实现，从而适用于一大类基于精度矩阵的近似方法。