Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

TL;DR

This paper introduces a scalable variational inference method for spatial Gaussian ICAR models, significantly reducing computational costs compared to traditional REML estimation.

Contribution

It develops a variational REML framework with proven convergence and exactness in Gaussian ICAR settings, improving efficiency over existing methods.

Findings

VREML outperforms MLE and INLA in empirical tests.

The ELBO converges monotonically, ensuring stable optimization.

The variational family is exact under Gaussian ICAR models.

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.