Estimation of Generalised linear mixed models (GLMM) including spatial Gaussian process models is often considered computationally impractical for even moderately sized datasets. In this article, we propose a fast Monte Carlo maximum likelihood (MCML) algorithm for the estimation of GLMMs. The algorithm is a stochastic Newton-Raphson method, which approximates the expected Hessian and gradient of the log-likelihood by drawing samples of the random effects. We propose a new stopping criterion for efficient termination and preventing long runs of sampling in the stationary post-convergence phase of the algorithm and discuss Monte Carlo sample size choice. We run a series of simulation comparisons of spatial statistical models alongside the popular integrated nested Laplacian approximation method and demonstrate potential for similar or improved estimator performance and reduced running times. We also consider scaling of the algorithms to large datasets and demonstrate a greater than 100-fold reduction in running times using modern GPU hardware to illustrate the feasibility of full maximum likelihood methods with big spatial datasets.

翻译：包含空间高斯过程模型在内的广义线性混合模型（GLMM）的估计，即使对于中等规模的数据集，也常被认为在计算上不切实际。本文提出一种用于估计GLMM的快速蒙特卡洛最大似然（MCML）算法。该算法是一种随机牛顿-拉弗森方法，通过对随机效应进行抽样来近似对数似然函数的期望海森矩阵和梯度。我们提出了一种新的停止准则，用于在算法的平稳收敛后阶段高效终止并防止长时间的抽样运行，并讨论了蒙特卡洛样本量的选择。我们针对空间统计模型进行了一系列与流行的集成嵌套拉普拉斯近似方法的模拟比较，结果表明该算法在估计器性能上具有相似或改进的潜力，同时减少了运行时间。我们还考虑了算法在大规模数据集上的扩展性，并展示了利用现代GPU硬件可将运行时间减少超过100倍，从而说明了全最大似然方法处理大型空间数据集的可行性。