Vecchia's approximate likelihood for Gaussian process parameters depends onhow the observations are ordered, which can be viewed as a deficiency becausethe exact likelihood is permutation-invariant. This article takes thealternative standpoint that the ordering of the observations can be tuned tosharpen the approximations. Advantageously chosen orderings can drasticallyimprove the approximations, and in fact, completely random orderings oftenproduce far more accurate approximations than default coordinate-basedorderings do. In addition to the permutation results, automatic methods forgrouping calculations of components of the approximation are introduced, havingthe result of simultaneously improving the quality of the approximation andreducing its computational burden. In common settings, reordering combined withgrouping reduces Kullback-Leibler divergence from the target model by a factorof 80 and computation time by a factor of 2 compared to ungroupedapproximations with default ordering. The claims are supported by theory andnumerical results with comparisons to other approximations, including taperedcovariances and stochastic partial differential equation approximations.Computational details are provided, including efficiently finding the orderingsand ordered nearest neighbors, and profiling out linear mean parameters andusing the approximations for prediction and conditional simulation. Anapplication to space-time satellite data is presented.
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