Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data
2026-04-08 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address the challenge of making fast and scalable estimates for spatial data using a method called Gaussian ICAR models. Traditional methods are slow because they need to repeatedly do complex matrix calculations. They propose a new approach called VREML, which uses a smart approximation to simplify the computations and speed up the estimation process. They also prove mathematically that their approximation is very accurate for these models and show through experiments that their method performs better than existing ones.
Gaussian ICARspatial datarestricted maximum likelihood (REML)variational inferenceevidence lower bound (ELBO)coordinate ascent algorithmprecision matrixrandom effectsvariance componentsINLA
Authors
Debjoy Thakur
Abstract
This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.