A scalable latent-free Vecchia approximation for Bayesian spatially varying coefficient models with censored data

Abstract Soil contamination by heavy metals, such as lead (Pb) and cadmium (Cd), poses a persistent threat to agricultural productivity and food safety, necessitating high-resolution spatial characterization for targeted remediation. Spatially varying coefficients (SVC) models offer a flexible framework for capturing non-stationary relationships between environmental predictors and contaminant concentrations across agricultural landscapes, overcoming the limitations of standard geostatistical models with external drift. While Bayesian SVC models, which treat coefficients as Gaussian processes, provide a rigorous inferential approach, their computational cost becomes prohibitive for large datasets due to the repeated inversion of dense covariance matrices within Markov chain Monte Carlo (MCMC) sampling. This study proposes an efficient Bayesian SVC framework that addresses this bottleneck through a novel latent-free Vecchia approximation, enabling scalable inference by directly evaluating the likelihood for censored observations without sampling high-dimensional latent fields. The approximation factorizes the observed-response likelihood and restricts each conditional set to previously ordered uncensored observations, so censored measurements enter through Gaussian tail probabilities rather than through sampled latent variables. The method is applied to a challenging case study of heavy metal contamination in agricultural experimental fields, characterized by substantial left-censoring due to detection limits and pronounced spatial heterogeneity. In simulation, the median relative log-likelihood error was at most 1.31% for $$M \ge 30$$ and censoring up to 50%, and the latent-free implementation required 0.87 h at $$M=45$$ compared with 4.20 h for the full latent model. The results indicate that the framework can recover spatially varying environmental effects and produce high-resolution risk summaries in the evaluated settings, suggesting a practical option for censored sensor-supported environmental monitoring when the stated limitations are considered.