858 Book Reviews ratio of two probability densities. Instead of using the log-likelihood ratio, one may use a power func- tion or some convex function of the likelihood ra- tio, a generalization due to A. Renyi and I. Csiszár in the 1960s. The exploitation of this generalized di- vergence measure to solve various statistical estima- tions and hypothesis testing problems is the subject of this beautifully produced book, which fills the gap of many recent developments since Kullback’s 1959 landmark Information Theory and Statistics. Chapter 1 gives a nice summary of various diver- gence measures, including those of J. Burbea and C. R. Rao. Chapter 2 discusses the issues in estimat- ing entropies from data whereas Chapters 3, 4 and 6 discuss applications in hypothesis testing. Chap- ter 5 covers minimum Φ-divergence estimators. A nice application of divergence measure is for com- positional and contingency data, which is discussed in Chapters 7 and 8. The issue of testing general populations is discussed in the final chapter. Fairly complete references are provided for readers who may want to explore the original sources. Reader- ship of the book should be statistical researchers who want to explore the boundaries of statistical inference paradigms and practitioners who are fre- quently faced with the needs to analyse categorical and discrete data. Z. Q. John Lu National Institute of Standards and Technology Gaithersburg Gaussian Markov Random Fields: Theory and Applications H. Rue and L. Held, 2005 Boca Raton, Chapman and Hall–CRC viii + 264 pp., $84.95 ISBN 1-584-88432-0 This book unifies and formalizes various strands in the emerging Bayesian literature on space–time smoothing. It considers inferential techniques and sampling methods that take advantage of the sparseness of the precision matrix Q (the inverse covariance matrix) in a class of Bayesian hier- archical models known as Gaussian graphical mo- dels, especially Gaussian Markov random fields (GMRFs). Whereas the covariance matrix is often of complicated form, for such models the preci- sion matrix typically includes model information in simple form: for autoregressive AR1 error models in time and conditional autoregressive CAR1 mod- els in space, 0s in precision matrices corresponding to conditional independences in such models. In Chapter 2 the authors consider more gener- ally the precision matrices for autoregressive mod- els in time, spatial conditional autoregressions and space–time GMRF models. In spatial applications, there is benefit in Markov chain Monte Carlo sam- pling terms to focus on simplifying even further the structure of Q, for instance by reordering of the nodes (areas). This leads to a minimal dia- gonal band structure in Q followed by Cholesky decomposition to take advantage of the band diagonalization. An example on pages 45–48 con- siders band Cholesky factorization for a map of Germany. Chapter 3 considers deficient rank precision ma- trices in improper or intrinsic GMRFs. It includes random-walk priors for equally and unequally spaced points (‘locations’) in time, and first- and higher order improper GMRFs on regular and irregular spatial lattices. Chapter 4 considers Markov chain Monte Carlo techniques in general, leading on to a consideration of block updating schemes that are of particular relevance in sampling for GMRF models (Knorr-Held and Rue, 2002). Applications in time series models (e.g. dynamic linear models type), semiparametric regression and binary regression with auxiliary variables are con- sidered. Univariate and multivariate count disease models are also discussed. Chapter 5 considers recent developments in GMRF methods, e.g. GMRF approximations in- stead of Gaussian fields in geostatistics. Appendix B contains more specific details for GMRF com- puting using the C routines in GMRFlib. This is a useful and up-to-date reference work in the area of space–time modelling and will complement related recent works such as those of Banerjee et al. (2004) and Waller and Gotway (2004). References Banerjee, S., Carlin, B. and Gelfand, A. (2004) Hierar- chical Modelling and Analysis for Spatial Data. Boca Raton: Chapman Hall–CRC. Knorr-Held, L. and Rue, H. (2002) On block updating in Markov random field models for disease mapping. Scand. J. Statist., 29, 597–614. Waller, L. and Gotway, C. (2004) Applied Spatial Statistics for Public Health Data. New York: Wiley. Peter Congdon Queen Mary College University of London Data Analysis of Asymmetric Structures T. Saito and H. Yadohisa, 2005 New York, Dekker viii + 258 pp., $99.95 ISBN 0-824-75398-4 What is asymmetry? The authors give no rigorous definition, and Chapter 1 presents it as one aspect of paired comparison data