Bayesian Inference for Non-Gaussian Simultaneous Autoregressive Models with Missing Data

TL;DR

This paper introduces new Bayesian spatial error models that handle non-Gaussian features like skewness and heavy tails, along with missing data, using variational Bayes and hybrid methods, validated on simulated and real datasets.

Contribution

The paper develops novel non-Gaussian SAR models with variational Bayes estimation, extending to missing data scenarios with a hybrid approach combining MCMC and VB.

Findings

Models effectively capture non-Gaussian spatial data characteristics.

Hybrid VB-MCMC method handles missing data under MNAR.

Proposed methods outperform standard Gaussian models in robustness.

Abstract

Standard simultaneous autoregressive (SAR) models typically assume normally distributed errors, an assumption often violated in real-world datasets that frequently exhibit non-normal, skewed, or heavy-tailed characteristics. New SAR models are proposed to capture these non-Gaussian features. The spatial error model (SEM), a widely used SAR-type model, is considered. Three novel SEMs are introduced, extending the standard Gaussian SEM. These extensions incorporate Student's $t$-distributed errors to accommodate heavy-tailed behaviour, one-to-one transformations of the response variable to address skewness, or a combination of both. Variational Bayes (VB) estimation methods are developed for these models, and the framework is further extended to handle missing response data under the missing not at random (MNAR) mechanism. Standard VB methods perform well with complete datasets; however, handling missing data requires a hybrid VB (HVB) approach, which integrates a Markov chain Monte Carlo (MCMC) sampler to generate missing values. The proposed VB methods are evaluated using both simulated and real-world datasets, demonstrating their robustness and effectiveness in dealing with non-Gaussian data and missing data in spatial models. Although the method is demonstrated using SAR models, the proposed model specifications and estimation approaches are widely applicable to various types of models for handling non-Gaussian data with missing values.