Variational Bayes inference for simultaneous autoregressive models with missing data

Abstract The simultaneous autoregressive (SAR) models are often used to analyse spatially correlated data. Markov chain Monte Carlo is one of the most widely used Bayesian methods for estimating the SAR models, 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. While the JVB method inaccurately estimates the posterior distributions of some SAR parameters and missing values, the standard HVB algorithm struggles to make accurate inferences when dealing with a large number of missing values. 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. While we demonstrate the method using SAR models, the approach has broad applicability to various models with missing data.