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Consistent Bayesian Spatial Domain Partitioning Using Predictive Spanning Tree Methods

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Nonparametric and PAC-Bayes Bayesian learning · 0 citations (OpenAlex)

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Proposes Spat-RPM, a spanning-tree-based Bayesian domain partition prior for spatially clustered regression that yields contiguous partitions usable for prediction, and proves posterior partition, cluster-number, parameter and prediction consistency under an infill-domain framework.

Problem

Bayesian spatial clustering models either cluster only finite observed locations (no prediction) or impose restrictive subregion shapes, and their posterior clustering consistency under infill-domain asymptotics is largely unexplored.

Approach

Discretize the domain into K^2 blocks, place a uniform spanning tree prior on the block grid graph, induce contiguous domain partitions by cutting tree edges with a Poisson prior on cluster number, and sample via birth/death/change/hyper MCMC moves; introduces a set-matching distance on domain partitions and derives consistency rates with order conditions on K and lambda.

Key results

• Over 100 repeats, posterior number of clusters matches the true k0 in most repeats when n>=30000, with underestimation for n<=10000.
• Against BSCC at n=4000, Spat-RPM concentrates on the true cluster number and has lower MedAE (mean difference -0.025, p<0.001) and CRPS (mean difference -0.0085, p=0.0074), with MCMC time 37s vs 71s.
• Sensitivity analysis on U-shape domain shows a U-shaped error curve in K and lambda; sigma^2 has negligible effect on partition error (mean errors 0.0294-0.0312).

Surprising findings

• A fixed, nondegenerate prior on the Poisson rate lambda is insufficient for cluster-number consistency; lambda must shrink with n.
• WAIC correlates poorly with true partition error when sigma^2=1, while CH index correlates poorly for larger sigma^2.

Limitations

• Theory restricted to Gaussian linear regression setting with fixed sigma^2 for proofs.
• Contraction rate n^{-1/2} log^... is slower than the parametric rate and simulations need n>30000 to see asymptotic behavior.
• Assumes K is pre-specified; a prior on K increases computational cost.

Experimental setup

Datasets: simulated spatial regression data on [0,1]^2 (CVT partitions) simulated U-shape domain data simulated areal data with hole Atlantic Ocean data

Benchmarks: simulated spatial regression under VDG regime U-shape domain simulation areal data simulation

Metrics: domain partition distance epsilon posterior number of clusters error regression coefficient estimation error prediction error CRPS MedAE computing time Calinski-Harabasz index WAIC

Baselines: BSCC

Builds on: uniform spanning tree prior Poisson-prior random partition models Bayesian spatially clustered coefficient models (BSCC) Zellner g-prior Prim's MST algorithm

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