Fast algorithms are developed for Bayesian analysis of Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) spatial random effects. To achieve computational speed-ups, first a result is proved on the equivalence between the use of an improper CAR prior with centering on the fly and the use of a sum-zero constrained ICAR prior. This equivalence result then provides the key insight for the algorithms, which are based on rewriting the hierarchical model in the spectral domain. The two novel algorithms are the Spectral Gibbs Sampler (SGS) and the Spectral Posterior Maximizer (SPM). Both algorithms are based on one single matrix spectral decomposition computation. After this computation, the SGS and SPM algorithms scale linearly with the sample size. The SGS algorithm is preferable for smaller sample sizes, whereas the SPM algorithm is preferable for sample sizes large enough for asymptotic calculations to provide good approximations. Because the matrix spectral decomposition needs to be computed only once, the SPM algorithm has computational advantages over algorithms based on sparse matrix factorizations (which need to be computed for each value of the random effects variance parameter) in situations when many models need to be fitted. Three simulation studies are performed: the first simulation study shows improved performance in computational speed in estimation of the SGS algorithm compared to an algorithm that uses the spectral decomposition of the precision matrix; the second simulation study shows that for model selection computations with 10 regressors and sample sizes varying from 49 to 3600, when compared to the current fastest state-of-the-art algorithm implemented in the R package INLA, SPM computations are 550 to 1825 times faster; the third simulation study shows that, when compared to default INLA settings, SGS and SPM combined with reference priors provide much more adequate uncertainty quantification. Finally, the application of the novel SGS and SPM algorithms is illustrated with a spatial regression study of county-level median household income for 3108 counties in the contiguous United States in 2017.

开发了具有固有条件自回归（ICAR）空间随机效应的高斯分层模型的贝叶斯分析快速算法。为了提高计算速度，首先证明了在以飞行为中心之前使用不适当的CAR和在先使用零和约束的ICAR之间的等效性。然后，该等效结果为算法提供了关键的见解，这些算法基于重写光谱域中的层次模型。两种新颖的算法是光谱吉布斯采样器（SGS）和光谱后验最大化器（SPM）。两种算法都基于一个单一矩阵频谱分解计算。经过此计算后，SGS和SPM算法与样本大小成线性比例关系。对于较小的样本量，首选SGS算法，而SPM算法更适合样本量足够大以进行渐近计算以提供良好的近似值。由于矩阵频谱分解仅需要计算一次，因此在需要拟合许多模型的情况下，SPM算法相对于基于稀疏矩阵分解的算法（需要针对随机效应方差参数的每个值进行计算）具有计算优势。 。进行了三个仿真研究：第一个仿真研究显示，与使用精度矩阵的频谱分解的算法相比，SGS算法的估计在计算速度上具有改进的性能；第二项仿真研究表明，对于使用10个回归变量和样本大小从49到3600不等的模型选择计算，与R包INLA中实现的当前最快的最新算法相比，SPM计算速度提高了550至1825倍；第三项仿真研究表明，与默认的INLA设置相比，SGS和SPM与参考先验相结合可提供更加充分的不确定性量化。最后，通过对2017年美国连续3108个县的县级家庭收入中位数进行空间回归研究，说明了新颖的SGS和SPM算法的应用。