The spatial error model (SEM) is a type of simultaneous autoregressive (SAR) model for analysing spatially correlated data. Markov chain Monte Carlo (MCMC) is one of the most widely used Bayesian methods for estimating SEM, but it has significant limitations when it comes to handling missing data in the response variable due to its high computational cost. Variational Bayes (VB) approximation offers an alternative solution to this problem. Two VB-based algorithms employing Gaussian variational approximation with factor covariance structure are presented, joint VB (JVB) and hybrid VB (HVB), suitable for both missing at random and not at random inference. When dealing with many missing values, the JVB is inaccurate, and the standard HVB algorithm struggles to achieve accurate inferences. Our modified versions of HVB enable accurate inference within a reasonable computational time, thus improving its performance. The performance of the VB methods is evaluated using simulated and real datasets.

翻译：空间误差模型（SEM）是一种用于分析空间相关数据的同步自回归（SAR）模型。马尔可夫链蒙特卡洛（MCMC）是估计SEM最广泛使用的贝叶斯方法之一，但由于其高昂的计算成本，在处理响应变量中的缺失数据时存在显著局限性。变分贝叶斯（VB）近似为这一问题提供了替代解决方案。本文提出了两种基于高斯变分近似并采用因子协方差结构的VB算法：联合变分贝叶斯（JVB）和混合变分贝叶斯（HVB），它们适用于随机缺失与非随机缺失的推断。当处理大量缺失值时，JVB方法不够精确，而标准的HVB算法难以实现准确的推断。我们对HVB的改进版本能够在合理的计算时间内实现精确推断，从而提升了其性能。通过模拟和真实数据集评估了这些VB方法的性能。