Contents

Preface

1 Introduction

1.1 Background

1.1.1 An introductory example

1.1.2 Conditional autoregressions

1.2 The scope of this monograph

1.2.1 Numerical methods for sparse matrices

1.2.2 Statistical inference in hierarchical models

1.3 Applications of GMRFs

2 Theory of Gaussian Markov random fields

2.1 Preliminaries

2.1.1 Matrices and vectors

2.1.2 Lattice and torus

2.1.3 General notation and abbreviations

2.1.4 Conditional independence

2.1.5 Undirected graphs

2.1.6 Symmetric positive-definite matrices

2.1.7 The normal distribution

2.2 Definition and basic properties of GMRFs

2.2.1 Definition

2.2.2 Markov properties of GMRFs

2.2.3 Conditional properties of GMRFs

2.2.4 Specification through full conditionals

2.2.5 Multivariate GMRFs_

2.3 Simulation from a GMRF

2.3.1 Some basic numerical linear algebra

2.3.2 Unconditional simulation of a GMRF

2.3.3 Conditional simulation of a GMRF

2.4 Numerical methods for sparse matrices

2.4.1 Factorizing a sparse matrix

2.4.2 Bandwidth reduction

2.5 A numerical case study of typical GMRFs

2.5.1 GMRF models in time

2.5.2 Spatial GMRF models

2.5.3 Spatiotemporal GMRF models

2.6 Stationary GMRFs_

2.6.1 Circulant matrices

2.6.2 Block-circulant matrices

2.6.3 GMRFs with circulant precision matrices

2.6.4 Toeplitz matrices and their approximations

2.6.5 Stationary GMRFs on infinite lattices

2.7 Parameterization of GMRFs_

2.7.1 The valid parameter space

2.7.2 Diagonal dominance

2.8 Bibliographic notes

3 Intrinsic Gaussian Markov random fields

3.1 Preliminaries

3.1.1 Some additional definitions

3.1.2 Forward differences

3.1.3 Polynomials

3.2 GMRFs under linear constraints

3.3 IGMRFs of first order

3.3.1 IGMRFs of first order on the line

3.3.2 IGMRFs of first order on lattices

3.4 IGMRFs of higher order

3.4.1 IGMRFs of higher order on the line

3.4.2 IGMRFs of higher order on regular lattices_

3.4.3 Nonpolynomial IGMRFs of higher order

3.5 Continuous-time random walks_

3.6 Bibliographic notes

4 Case studies in hierarchical modeling

4.1 MCMC for hierarchical GMRF models

4.1.1 A brief introduction to MCMC

4.1.2 Blocking strategies

4.2 Normal response models

4.2.1 Example: Drivers data

4.2.2 Example: Munich rental guide

4.3 Auxiliary variable models

4.3.1 Scale mixtures of normals

4.3.2 Hierarchical-t formulations

4.3.3 Binary regression models

4.3.4 Example: Tokyo rainfall data

4.3.5 Example: Mapping cancer incidence

4.4 Nonnormal response models

4.4.1 The GMRF approximation

4.4.2 Example: Joint disease mapping

4.5 Bibliographic notes

5 Approximation techniques

5.1 GMRFs as approximations to Gaussian fields

5.1.1 Gaussian fields

5.1.2 Fitting GMRFs to Gaussian fields

5.1.3 Results

5.1.4 Regular lattices and boundary conditions

5.1.5 Example: Swiss rainfall data

5.2 Approximating hidden GMRFs

5.2.1 Constructing non-Gaussian approximations

5.2.2 Example: A stochastic volatility model

5.2.3 Example: Reanalyzing Tokyo rainfall data

5.3 Bibliographic notes

Appendices

A Common distributions

B The library GMRFLib

B.1 The graph object and the function Qfunc

B.2 Sampling from a GMRF

B.3 Implementing block-updating algorithms for hierarchical
GMRF models