MCMC-Free Uncertainty Quantification for Deep Generative Inference for Spatially Varying Coefficient Models at Scale

2026-09-28

AI Summary

This study addresses the scalability bottleneck of varying coefficient models for large-scale spatial data caused by Markov Chain Monte Carlo (MCMC) sampling, proposing the GeoVAE framework. Departing from conventional Gaussian process covariance representations, this method pioneers dedicated autoencoders for individual coefficients and introduces a hierarchical synthesis layer. By explicitly modeling cross-coefficient dependencies through a hierarchical deep generative model, GeoVAE jointly estimates spatially varying coefficients and enables efficient uncertainty quantification without MCMC. The proposed framework significantly reduces computational costs while preserving its capacity to recover complex spatial patterns, thereby offering an efficient deep learning solution for large-scale spatial analysis.

Abstract

Varying coefficient (VC) regression models have become indispensable tools in spatial data analysis, providing unmatched flexibility in capturing complex, nonlinear relationships and spatially-varying effects of predictors on responses. Although hierarchical Bayesian approaches offer a rigorous probabilistic framework for uncertainty quantification in VC modeling, their practical application to large-scale spatial datasets remains severely hindered by the scalability limitations of Markov chain Monte Carlo (MCMC) algorithms. In response, the past decade has seen substantial advances in developing more efficient hierarchical Bayesian VC models, primarily through substituting traditional Gaussian processes (GP) with computationally efficient stochastic surrogates for estimating unknown coefficient functions. This article introduces a fundamentally different approach: the Geostatistical Variational Auto-Encoder (GeoVAE), a hierarchical deep generative framework built specifically for joint estimation of multiple spatially varying coefficient functions. GeoVAE departs from both classical GP-based Bayesian models and standard variational auto-encoders (VAEs) in two key ways. First, it constructs a coefficient-specific auto-encoder for each coefficient function, allowing each to capture its own spatial resolution and smoothness. Second, a hierarchical synthesis layer integrates information across these auto-encoders to a shared auto-encoder, explicitly modeling cross-coefficient dependencies arising from the shared spatial domain. This design enables GeoVAE to recover complex spatial patterns at computational cost that scales favorably with sample size, without MCMC or explicit GP covariance representations. We characterize the advantages and limitations of GeoVAE relative to hierarchical Bayesian spatial models, demonstrating its strong potential for large-scale spatial analysis.

Problem

• Uncertainty Quantification
• Varying Coefficient Models
• Spatial Data Analysis
• Scalability
• MCMC

Innovation

• Geostatistical Variational Auto-Encoder
• Spatially Varying Coefficient Models
• MCMC-Free Uncertainty Quantification
• Hierarchical Deep Generative Framework
• Scalable Spatial Analysis