Latent Gaussian process (GP) models are flexible probabilistic non-parametric function models. Vecchia approximations are accurate approximations for GPs to overcome computational bottlenecks for large data, and the Laplace approximation is a fast method with asymptotic convergence guarantees to approximate marginal likelihoods and posterior predictive distributions for non-Gaussian likelihoods. Unfortunately, the computational complexity of combined Vecchia-Laplace approximations grows faster than linearly in the sample size when used in combination with direct solver methods such as the Cholesky decomposition. Computations with Vecchia-Laplace approximations can thus become prohibitively slow precisely when the approximations are usually the most accurate, i.e., on large data sets. In this article, we present iterative methods to overcome this drawback. Among other things, we introduce and analyze several preconditioners, derive new convergence results, and propose novel methods for accurately approximating predictive variances. We analyze our proposed methods theoretically and in experiments with simulated and real-world data. In particular, we obtain a speed-up of an order of magnitude compared to Cholesky-based calculations and a threefold increase in prediction accuracy in terms of the continuous ranked probability score compared to a state-of-the-art method on a large satellite data set. All methods are implemented in a free C++ software library with high-level Python and R packages.

翻译：潜高斯过程模型是灵活的概率非参数函数模型。Vecchia近似是高斯过程的一种精确近似方法，用于克服大数据计算瓶颈；而Laplace近似是一种快速方法，具有渐近收敛保证，可用于近似非高斯似然下的边缘似然和后验预测分布。然而，当结合Cholesky分解等直接求解器使用时，组合Vecchia-Laplace近似的计算复杂度随样本量增长的速度快于线性。因此，在近似方法通常最精确（即处理大型数据集）时，Vecchia-Laplace近似的计算可能变得极其缓慢。本文提出迭代方法来克服这一缺陷。我们引入并分析了多种预条件子，推导了新的收敛性结果，并提出了精确近似预测方差的新方法。我们通过理论分析和模拟及真实数据实验对所提方法进行验证。特别地，在大型卫星数据集上，相比基于Cholesky分解的计算方法，我们获得了数量级的加速；相比最先进方法，在连续分级概率评分指标下预测精度提升三倍。所有方法均已在开源C++软件库中实现，并提供高级Python和R语言接口。