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**Gaussian Markov random fields. Theory and applications.**
*(English)*
Zbl 1093.60003

Monographs on Statistics and Applied Probability 104. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-432-0/hbk; 978-0-203-49202-4/ebook). xi, 263 p. (2005).

This monograph provides a well-written and comprehensive account of the main properties of Gaussian Markov random fields (GMRFs) and intrinsic GMRFs, and their connection to sparse matrices, computational issues and applications. Numerical methods for sparse matrices are presented and used for simulations of GMRFs and evaluation of the log density of a GMRF. These methods are used for fast and reliable Bayesian inference in hierarchical GMRFs models via Markov chain Monte Carlo simulations. The developed theory is exemplified and supported by detailed case studies from various areas ranging from structural time series, over scale mixtures of normals, to binary regression models in spatiotemporal statistics. The monograph is accompanied by the online and open source library “GMRFLib”, and C-code implementations of simulations of GMRFs and of block-updating algorithms for hierarchical GMRF models are included. Thus, the monograph both gives a theoretical overview and provides a toolbox for practical applications. Below we provide a more detailed description of the five chapters and two appendices in the 263 pages long monograph.

Chapter 1 provides introduction, motivations and a comprehensive list of applications with references. Chapter 2 provides details on the theory of GMRFs and their connection to sparse matrices. Simulation from GMRFs and numerical methods for sparse matrices are presented and discussed. Chapter 3 extends the theory for GMRFs to intrinsic GMRFs (IGMRFs). IGMRFs are improper, i.e. have precision matrix not of full rank, and can be conceived as GMRF under linear constraints. In particular, the first and second order random walk and the relation to \(k\)-fold integrated Wiener processes are discussed. Furthermore, stencils for higher order IGMRFs on the regular 2D-lattice are presented.

Chapter 4 is devoted to case studies in hierarchical modeling and inference. The chapter begins with a brief introduction to MCMC methods and blocking strategies. First applications include normal response models, but employing auxiliary variables the applications are extended to scale mixtures of normals and binary regression models. Finally, it is discussed how GMRFs can be employed as Metropolis-Hastings proposals in general non-normal response models. In particular, Poisson response is considered. All these considerations are accompanied by detailed case studies on real data. Chapter 5 is devoted to more general considerations on two approximation methods via GMRFs. Firstly, the approximation of Gaussian fields is investigated. Secondly, the approximation of hidden GMRFs is investigated. Both situations are supported by case studies on real data. Chapters 2 to 5 are all concluded by bibliographic notes. Appendix A provides a brief overview of common distributions. Appendix B provides a short introduction to the “GMRFLib” library, including C-code with examples. The monograph concludes with 18 pages of references, and author and subject indexes.

Chapter 1 provides introduction, motivations and a comprehensive list of applications with references. Chapter 2 provides details on the theory of GMRFs and their connection to sparse matrices. Simulation from GMRFs and numerical methods for sparse matrices are presented and discussed. Chapter 3 extends the theory for GMRFs to intrinsic GMRFs (IGMRFs). IGMRFs are improper, i.e. have precision matrix not of full rank, and can be conceived as GMRF under linear constraints. In particular, the first and second order random walk and the relation to \(k\)-fold integrated Wiener processes are discussed. Furthermore, stencils for higher order IGMRFs on the regular 2D-lattice are presented.

Chapter 4 is devoted to case studies in hierarchical modeling and inference. The chapter begins with a brief introduction to MCMC methods and blocking strategies. First applications include normal response models, but employing auxiliary variables the applications are extended to scale mixtures of normals and binary regression models. Finally, it is discussed how GMRFs can be employed as Metropolis-Hastings proposals in general non-normal response models. In particular, Poisson response is considered. All these considerations are accompanied by detailed case studies on real data. Chapter 5 is devoted to more general considerations on two approximation methods via GMRFs. Firstly, the approximation of Gaussian fields is investigated. Secondly, the approximation of hidden GMRFs is investigated. Both situations are supported by case studies on real data. Chapters 2 to 5 are all concluded by bibliographic notes. Appendix A provides a brief overview of common distributions. Appendix B provides a short introduction to the “GMRFLib” library, including C-code with examples. The monograph concludes with 18 pages of references, and author and subject indexes.

Reviewer: Bo Markussen (Kopenhagen)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G60 | Random fields |

60G15 | Gaussian processes |

62F15 | Bayesian inference |

62M40 | Random fields; image analysis |

65C05 | Monte Carlo methods |

65F50 | Computational methods for sparse matrices |