Theory of Gaussian Markov random fields
Havard Rue, Leonhard Held · 2005
In this chapter, we will present the basic properties of a GMRF. As a GMRF is normal, all results valid for a normal distribution are also valid for a GMRF. However, in order to apply GMRFs in Bayesian hierarchical models, we need to sample from GMRFs and to compute certain properties of GMRFs under various conditions. What makes GMRFs extremely useful in practice, is that the things we often need to compute are particularly fast to compute for a GMRF. The key is naturally the sparseness of the precision matrix and the structure of its nonzero terms. It will be useful to represent GMRFs on a graph representing the nonzero pattern of the precision matrix. This representation serves two purposes. First, it will provide a unified way of interpreting and understanding a GMRF through conditional independence, either for a GMRF in time, on a lattice, or on some more general structure. Secondly, this representation will also provide a unified way to actually compute various properties for a GMRF and to generate samples from it, by using numerical methods for sparse matrices.