Consistent Bayesian Spatial Domain Partitioning Using Predictive Spanning Tree Methods

Bayesian model-based spatial clustering methods are widely used for their flexibility in estimating latent clusters with an unknown number of clusters while accounting for spatial proximity. Many existing methods are designed for clustering finite spatial units, limiting their ability to make predictions, or may impose restrictive geometric constraints on the shapes of subregions. Furthermore, the posterior clustering consistency theory of spatial clustering models remains largely unexplored in the literature. In this study, we propose a Spatial Domain Random Partition Model (Spat-RPM) and demonstrate its application for spatially clustered regression, which extends spanning tree-based Bayesian spatial clustering by partitioning the spatial domain into disjoint blocks and using spanning tree cuts to induce contiguous domain partitions. Under an infill-domain asymptotic framework, we introduce a new distance metric to study the posterior concentration of domain partitions. We show that Spat-RPM achieves a consistent estimation of domain partitions, including the number of clusters (which may go to infinity), and derive posterior concentration rates for partition, parameter, and prediction. We also establish conditions on the hyperparameters to achieve consistency, offering important practical guidance for hyperparameter selection. Finally, we examine the asymptotic properties of our model through simulation studies and apply it to Atlantic Ocean data.