Since their introduction in statistics through the seminal works of Julian Besag, Gaussian Markov random fields have become central to spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. Specified by a set of conditional distributions, these Markov random fields provide a very rich and flexible class of spatial processes, and their adaptability to fast statistical calculations, including those based on Markov chain Monte Carlo computations, makes them very attractive to statisticians. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. In this talk, I will briefly discuss the scaling limit of lattice-based Gaussian Markov random fields, namely,  the de Wijs process that originates in the famous work of George Matheron on gold mines in South Africa. I will then explore how this continuum limit connection holds out further possibilities to fit a wide range of new continuum models by using lattice-based random fields. The main focus of the talk will be on matrix-free computation for these models. In particular, for spatial mixed linear models, I will present frequentist residual maximum likelihood inference via matrix-free h-likelihood computation. I will draw applications both from areal-unit and point-referenced spatial data.

The work resulted from collaborations with (late) Julian Besag, and former  PhD students Somak Dutta and Chunxiao Wang.