Parallel exact sampling and evaluation of Gaussian Markov random fields

Markov chain Monte Carlo algorithms are computationally expensive for large models. Especially, the so-called one-block Metropolis–Hastings (M–H) algorithm demands large computational resources, and parallel computing seems appealing. A parallel one-block M–H algorithm for latent Gaussian Markov random field (GMRF) models is introduced. Important parts of this algorithm are parallel exact sampling and evaluation of GMRFs. Parallelisation is achieved with parallel algorithms from linear algebra for sparse symmetric positive definite matrices. The parallel GMRF sampler is tested for GMRFs on lattices and irregular graphs, and gives both good speed-up and good scalability. The parallel one-block M–H algorithm is used to make inference for a geostatistical GMRF model with a latent spatial field of 31,500 variables.