Professor Håvard Rue
Department of Mathematical Sciences
Norwegian University of Science and Technology
Personal website: http://www.math.ntnu.no/~hrue
12:00 – 14:00, Classroom 0
1. Introduction and outline of the course, background.
2. An introduction to Gaussian Markov random fields; part I
17:00 – 19:00, Classroom 10
3. An introduction to Gaussian Markov random fields; part II
4. An introduction to Bayesian statistics
Day 2 (Thursday, 11/11/2010)
10:00 – 12:00, Classroom 10
5. Integrated nested Laplace approximations; part I
6. Integrated nested Laplace approximations; part II
16:00 – 19:00, Computer lab. I2
7. Examples and case-studies in R
8. Examples and case-studies in R
9. Extensions: Spatial models and Log-Gaussian Cox-processes.
Håvard Rue is professor in statistics at the Department of Mathematical Sciences, Norwegian University of Science and Technology. His research interest includes Bayesian computing and spatial statistics, which is summarised through R-INLA package, see http://www.r-inla.org/. He has been an associate editor for JRSS series-B, Scandinavian Journal of Statistics, Statistic Surveys, Annals of Statistics and Environmetrics. His main research interest has been Gaussian Markov random fields (GMRF) models, and with Leonhard Held he has written a monograph on the subject published by Chapman & Hall. GMRFs is also a main ingredient doing (fast and accurate) approximate Bayesian analysis for latent Gaussian models using integrated nested Laplace approximations (INLA), which is published as a discussion paper for JRSS series B 2009 co-authored with S.Martino and N.Chopin. The webpapge http://www.r-inla.org/ provides an R-interface to the INLA methodology. The recent research interest is taking GMRFs into geostatistics using stochastic partial differential equations as the bridge, which provides an explicit link between certain Gaussian fields and GMRFs in triangulated lattices.
In these lectures, I will discuss approximate Bayesian inference for a class of models named `latent Gaussian models' (LGM). LGM's are perhaps the most commonly used class of models in statistical applications. It includes, among others, most of (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models.
The concept of LGM is intended for the modeling stage, but turns out to be extremely usefull when doing inference as we can treat models listed above in a unified way and using the “same” algorithm and software tool. Our approach to (approximate) Bayesian inference, is to use integrated nested Laplace approximations (INLA). Using this new tool, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.
In these lectures I will introduce the required background and theory for understanding INLA, including details on Gaussian Markov random fields and fast computations of those using sparse matrix algorithms. I will end these lectures illustrating INLA on a range of examples in R (see http://www.r-inla.org/).
Rosa M. Crujeiras Casais
Dpt. of Statistics and Operations Research, Faculty of Mathematics
University of Santiago de Compostela