Sampling Strategies for Fast Updating of Gaussian Markov Random Fields

Gaussian Markov random fields (GMRFs) are popular for modeling dependence in large areal datasets due to their ease of interpretation and computational convenience afforded by the sparse precision matrices needed for random variable generation. Typically in Bayesian computation, GMRFs are updated jointly in a block Gibbs sampler or componentwise in a single-site sampler via the full conditional distributions. The former approach can speed convergence by updating correlated variables all at once, while the latter avoids solving large matrices. We consider a sampling approach in which the underlying graph can be cut so that conditionally independent sites are updated simultaneously. This algorithm allows a practitioner to parallelize updates of subsets of locations or to take advantage of “vectorized” calculations in a high-level language such as R. Through both simulated and real data, we demonstrate computational savings that can be achieved versus both single-site and block updating, regardless of whether the data are on a regular or an irregular lattice. The approach provides a good compromise between statistical and computational efficiency and is accessible to statisticians without expertise in numerical analysis or advanced computing.

高斯马尔可夫随机场（Gaussian Markov random fields，GMRFs）因其易于解释和计算便利性而广受欢迎，这种便利性源于随机变量生成所需的稀疏精度矩阵。在贝叶斯计算中，GMRFs通常通过块吉布斯采样器联合更新，或在单站点采样器中逐分量更新，通过完全条件分布实现。前者通过一次更新相关变量来加速收敛，而后者避免了求解大型矩阵。我们考虑了一种采样方法，其中底层图可以被切割，以便条件独立站点可以同时更新。此算法允许从业者并行化子集位置的更新，或利用高级语言如R中的“向量化”计算。通过模拟和真实数据，我们展示了与单站点和块更新相比可实现的计算节省，无论数据是否位于规则或不规则的格子上。这种方法在统计效率和计算效率之间提供了良好的折衷，并且对没有数值分析或高级计算专业知识的数据统计学家来说是可访问的。