  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.
