Bayesian inference for the automultinomial model with an application to landcover data

TL;DR

The paper tackles modeling multicategory lattice data with spatial dependence by introducing the automultinomial framework, an extension of the Potts model that incorporates covariates while preserving a single spatial correlation parameter. Because the likelihood involves an intractable normalizing function $Z(oldsymbol{eta}, oldsymbol{ abla})$, the authors develop Bayesian inference via the Double Metropolis–Hastings algorithm and accompany it with the Approximate Curvature Diagnostic to assess convergence. Through simulation studies with CAR random effects and mixtures of Gaussian processes, they demonstrate the model's flexibility across a range of spatial dependencies and provide practical guidelines for tuning and diagnostics. An application to Southeast Asian land-cover data shows the approach can recover major spatial patterns and covariate associations, offering a practical tool for scientists despite challenges in class imbalance and local misfit. Overall, the work delivers a versatile Bayesian framework for multicategory areal data with actionable recommendations for specification and computation.

Abstract

Multicategory lattice data arise in a wide variety of disciplines such as image analysis, biology, and forestry. We consider modeling such data with the automultinomial model, which can be viewed as a natural extension of the autologistic model to multicategory responses, or equivalently as an extension of the Potts model that incorporates covariate information into a pure-intercept model. The automultinomial model has the advantage of having a unique parameter that controls the spatial correlation. However, the model's likelihood involves an intractable normalizing function of the model parameters that poses serious computational problems for likelihood-based inference. We address this difficulty by performing Bayesian inference through the Double-Metropolis Hastings algorithm, and implement diagnostics to assess the convergence to the target posterior distribution. Through simulation studies and an application to land cover data, we find that the automultinomial model is flexible across a wide range of spatial correlations while maintaining a relatively simple specification. For large data sets we find it also has advantages over spatial generalized linear mixed models. To make this model practical for scientists, we provide recommendations for its specification and computational implementation.

Figures (6)

• Figure 1: Simulated arrangement from hierarchical model with CAR random effects (a) and predicted arrangement under automultinomial model (b)
• Figure 2: Simulated arrangement from mixture of Gaussian Processes
• Figure 3: Simulated arrangement from mixture of Gaussian Processes (a) and predicted arrangement under automultinomial model (b)
• Figure 4: Land cover in the region of study
• Figure 5: Covariates in the region of study