Think continuous: Markovian Gaussian models in spatial statistics

Gaussian Markov random fields (GMRFs) are frequently used as computationally efficient models in spatial statistics. Unfortunately, it has traditionally been difficult to link GMRFs with the more traditional Gaussian random field models, as the Markov property is difficult to deploy in continuous space. Following the pioneering work of Lindgren et al. (2011), we expound on the link between Markovian Gaussian random fields and GMRFs. In particular, we discuss the theoretical and practical aspects of fast computation with continuously specified Markovian Gaussian random fields, as well as the clear advantages they offer in terms of clear, parsimonious, and interpretable models of anisotropy and non-stationarity.