Norwegian University of Science and Technology - Department of Mathematical Sciences
Gaussian Markov Random Fields (GMRFs) has been around for a long time; however, it is first in the recent years that its computational benefits in Bayesian inference has become clear. In this talk, I'll discuss two related problems which involves GMRFs. The first is the problem of constructing Gaussian fields on triangulated manifolds. By viewing this as finding the solution of a stochastic partial differential equation (SPDE), the GMRFs appear as the solutions when solving the SPDE using the "finite element" approach. A special case provides some new results on the connection between GMRFs and Gaussian fields with a Matern covariance function; we obtain an explicit parametrisation for a GMRF on a regular lattice, which defines (in a weak sense) a Gaussian field with a Matern covariance function. The second theme is the use of GMRFs for doing approximate Bayesian inference for latent Gaussian models; this approach provides posterior marginals that totally outperform natural MCMC alternatives in both accuracy and speed.