(\[ascore\]) is that if at any point one wants to obtain a better approximation to the score function, it suffices to consider additional $U_j$’s in (\[ascore\]). However, how exactly to do this if using the dependent sampling scheme for the $U_j$’s in Section \[sec4\] is not so obvious.

Since this stochastic approach provides only an approximation to the MLE, one must compare it with other possible approximations to the MLE. Many such approaches exist, including spectral methods, low-rank approximations, covariance tapering and those based on some form of composite likelihood. All these methods involve computing the likelihood itself and not just its gradient, and thus all share this advantage over solving (\[ascore\]). Note that one can use randomized algorithms to approximate $\log\det K$ and thus approximate the loglikelihood directly \[@zhangY\]. However, this approximation requires first taking a power series expansion of $K$ and then applying the randomization trick to each term in the truncated power series; the examples presented by @zhangY show that the approach does not generally provide a good approximation to the loglikelihood. Since the accuracy of the power series approximation to $\log\det K$ depends on the condition number of $K$, some of the filtering ideas described by 