# Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

arXiv: https://arxiv.org/abs/2604.07635 · HTML: https://mlsanity.com/paper/2604.07635/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 11 of 11)

- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Gaussian variational approximation with a factor covariance structure](https://mlsanity.com/paper/1701.03208/) (2017)
- [On the Convergence of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2306.01122/) (2023)
- [Gaussian variational approximation with sparse precision matrices](https://mlsanity.com/paper/1605.05622/) (2016)
- [Variational Bayes with Intractable Likelihood](https://mlsanity.com/paper/1503.08621/) (2015)
- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Fast and Accurate Variational Inference for Models with Many Latent Variables](https://mlsanity.com/paper/2005.07430/) (2020)
- [Scalable logistic regression with crossed random effects](https://mlsanity.com/paper/2105.13747/) (2021)
- [Fast Bayesian Estimation of Spatial Count Data Models](https://mlsanity.com/paper/2007.03681/) (2020)
- [Computationally Efficient Bayesian Unit-Level Models for Non-Gaussian Data Under Informative Sampling](https://mlsanity.com/paper/2009.05642/) (2020)
- [A Scalable Variational Bayes Approach to Fit High-dimensional Spatial Generalized Linear Mixed Models](https://mlsanity.com/paper/2402.15705/) (2024)

## Abstract (arXiv)

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.