Markov chain Monte Carlo (MCMC) allows one to generate dependent replicates from a posterior distribution for effectively any Bayesian hierarchical model. However, MCMC can produce a significant computational burden. This motivates us to consider finding expressions of the posterior distribution that are computationally straightforward to obtain independent replicates from directly. We focus on a broad class of Bayesian latent Gaussian process (LGP) models that allow for spatially dependent data. First, we derive a new class of distributions we refer to as the generalized conjugate multivariate (GCM) distribution. The GCM distribution's theoretical development is similar to that of the CM distribution with two main differences; namely, (1) the GCM allows for latent Gaussian process assumptions, and (2) the GCM explicitly accounts for hyperparameters through marginalization. The development of GCM is needed to obtain independent replicates directly from the exact posterior distribution, which has an efficient projection/regression form. Hence, we refer to our method as Exact Posterior Regression (EPR). Illustrative examples are provided including simulation studies for weakly stationary spatial processes and spatial basis function expansions. An additional analysis of poverty incidence data from the U.S. Census Bureau's American Community Survey (ACS) using a conditional autoregressive model is presented.

翻译：马尔可夫链蒙特卡洛方法（MCMC）能够从任何贝叶斯层次模型的后验分布生成相依的重复样本。然而，MCMC会产生显著的计算负担。这促使我们探索在计算上能够直接获取后验分布独立重复样本的表达形式。我们聚焦于一类允许空间相依数据的广义贝叶斯潜在高斯过程（LGP）模型。首先，我们推导出一类新的分布，称为广义共轭多元（GCM）分布。GCM分布的理论发展与CM分布相似，但存在两个主要区别：（1）GCM允许潜在高斯过程假设；（2）GCM通过边缘化显式处理超参数。发展GCM分布的目的是直接从精确后验分布中获取独立重复样本，该后验分布具有高效的投影/回归形式。因此，我们将该方法称为精确后验回归（EPR）。我们提供了说明性示例，包括弱平稳空间过程的模拟研究和空间基函数展开。此外，还利用条件自回归模型对美国人口普查局美国社区调查（ACS）的贫困发生率数据进行了额外分析。