Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data
This work addresses scalability issues in spatial data analysis for researchers and practitioners, representing an incremental improvement over existing methods.
The paper tackles the computational cost of restricted maximum likelihood estimation for spatial data with Gaussian ICAR structures by proposing a variational restricted maximum likelihood framework, which approximates the marginal likelihood and achieves efficient estimation with theoretically exact variational approximation under Gaussian settings, empirically outperforming MLE and INLA.
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.