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Numerical Bayesian Methods Applied to SignalProcessingKurt S. Riedel aa New York UniversityPublished online: 12 Mar 2012.

To cite this article: Kurt S. Riedel (1997) Numerical Bayesian Methods Applied to Signal Processing, Technometrics, 39:1,106-106

To link to this article: http://dx.doi.org/10.1080/00401706.1997.10485453

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106 BOOK REVIEWS

source code listing for S-Plus functions to simulate fractional Gaussian noise and fractional ARIMA processes and to calculate many of the esti- mates discussed in Chapters 5-l I, Chapter 12 also contains listing of the data for the five examples presented in Chapter 1.

Overall, the book provides a solid background in long-memory pro- cesses and a wide-ranging look at the current state of research on statisti- cal inference for these processes. It appears to be well suited as a primary source both for researchers seeking to expand their knowledge of stochas- tic processes into this area and for researchers currently working in this area. Note, however, that, although the book explores its five examples in detail, it contains no exercises, limiting its usefulness as a textbook for courses in advanced stochastic processes and time series.

Jeffrey GLOSUP Lawrence Livermore National Laboratory

REFERENCES

Cox. D. R. (1984), “Long-Range Dependence: A Review,” in Statistics: An Appraisal, Proceedings of the 50th Anniversary Conference, Iowa State Statistical Laboratory eds. H. A. David, and H. T. David, Ames: The Iowa State University Press, pp. 55-74.

Harvey, A. C. (1993), Time Series Models (2nd ed.), Cambridge, MA: MIT Press.

Numerical Bayesian Methods Applied to Signal Pro- cessing, by Joseph J. K. 0 RUANAIDH and William J. FITZGERALD, New York: Springer-Verlag, 1996, xiv + 244 pp., $49.95.

The book focuses on using Markov-chain Monte Carlo (MCMC) meth- ods on Bayesian estimation problems. The authors are well-meaning en- gineers who apply MCMC methods. Many topics are treated on a surface level, but no topic is covered in detail. The authors’ style is to insert sev- eral paragraphs of motivational discussion between equations rather than writing down a shorter mathematical derivation.

Chapters 1 and 2 describe the authors’ philosophy, introduce notation, and perform the standard reduction of Gaussian likelihood in a linear model. Personally, I recommend learning this material from Box and Tiao (1973) instead. Chapter 3 begins with a review of material in advanced calculus and introductory numerical analysis. Most of this material was given by Press, Teukolsky, Vetterling, and Flannery (1992). In most cases, the authors have paraphrased Press et al. in a wordy, imprecise fashion. Amusingly, the authors credit the partitioned matrix inversion identity to Press et al.

Chapter 4 describes MCMC methods at an elementary level. For deeper understanding, the reader should consult the many excellent tutorial and review articles, such as those by Besag, Green, Higdon, and Mengerson (1995), Casella and George (1992) George, Makov, and Smith (1994), Smith and Gelfand (1992), and Smith and Roberts (1993).

The examples in Chapters 5-7 are described in detail, allowing the reader to see how the general theory is applied in specific cases. Chapter 5 applies the MCMC method to change-point detection. The analysis differs from the frequentist perspective (Basseville and Nikiforov 1993) through a logarithmic prior on the variance.

Chapter 6 examines autoregressive parameters estimation and signal re- covery with missing data. The Gibbs sampler, the estimated maximum likelihood method, and the maximum likelihood method are described and applied to real examples. The authors incorrectly assume that the Gaus- sian likelihood is the sum of squares of the residual errors. (They use the likelihood conditional on the first p observations instead of the full likeli- hood.) A treatment of the true likelihood of an autoregressive process may be found in the work of Brockwell and Davis (1991). The data fits of the three methods are plotted and judged for aesthetic quality, but there are no statistical comparisons.

Chapter 7 tries to estimate a fourth-order polynomial using MCMC methods. Oddly, the authors do not use orthonormal polynomials. The results appear to be so bad that only one of two inferences is possible: Either something is bad in the authors’ formulation or one should not use MCMC methods for regression problems with strongly correlated basis functions.

The book may be useful for physicists and engineers who wish to modify the detailed examples to tit their particular cases. Most statisticians will be better off reading one or more of the review/tutorial articles I have cited, supplemented when necessary by an introductory numerical analysis textbook.

Kurt S. RIEDEL New York University

REFERENCES

Basseville, M., and Nikiforov, I. V. (1993), Detection qf Abrupt Changes; Theory and Application, Englewood Cliffs, NJ: PTR Prentice-Hall.

Besag, J., Green, P., Higdon, D., and Mengersen, K. (1995), “Bayesian Computation and Stochastic Systems,” Statistical Science, 10, 346.

Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, New York, Addison-Wesley.

Brockwell, P. J., and Davis, R. A. (199 l), Time Series: Theory and Models (Springer Series in Statistics), New York: Springer-Verlag.

Casella, G., and George, E. I. (1992), “Explaining the Gibbs Sampler,” The American Statistician, 46, 167-174.

George, E. I., Makov, U. E., and Smith, A. F. M. (1994), “Fully Bayesian Hierarchical Analysis for Exponential Families via Monte Carlo Com- putation,” in Aspects of Uncertainty, eds. P. R. Freeman and A. F. M. Smith, New York: Wiley.

Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992) Numerical Recipes in C, New York: Cambridge University Press.

Smith, A. F. M., and Gelfand, A. E. (1992), “Bayesian Statistics Without Tears: A Sampling-Resampling Perspective,” The American Stafisricinn, 46, 84-88.

Smith, A. F. M., and Roberts, G. 0. (1993), “Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods,” Journal of the Royal Statistical Society, Ser. B, 55, 3-23.

Fundamentals of Applied Statistics and Surveys, by David B. ORR, New York: Chapman & Hall, 1995, xxv + 346 pp., $39.95 (softcover).

This book is good for anyone wanting an intuitive understanding of statistics. The target audience as stated by the author is noncareer statis- ticians. This book emphasizes the statistical concepts over the “how to’s” of statistics. If you are looking for formulas, this book is not for you. but, if you want to know why you should care about correlation or hypothesis testing, it addresses these issues.

The topics covered in this book include introduction to numbers and statistics, graphical displays, averages, variability, correlation, regression, analysis of variance (ANOVA), and the basics of surveys. Only the most basic calculations are illustrated for each topic. The author focuses his effort on conceptual understanding and the reasons the reader should care about these statistical topics. Simple examples are often used to reinforce mathematical concepts in lieu of mathematical proofs. These examples encourage readers to develop their own intuitiveness about the topic if it is not already present.

The book is missing some topics that are present in most introductory statistics books. These include two-sample tests of hypotheses, sample size, nonparametrics, and two-way ANOVA. The book does, however, in- clude several concepts that are not always in introductory statistics books. These include power, moments, true zero, outliers, and some interesting definitions of statistical terms. The concepts presented are not new, nor are the methods used to address them. Ideas in this book mirror works by Freedman, Pisani, and Purves (1978) for concepts over formulas or Huff (1954) for how graphs can be misleading. The absence of exercises would make it more challenging to use this as a sole textbook for a statistics class. The slant of the book is toward the social sciences.

I like the way the author uses the reader’s intuition through simple examples to aid in the understanding process. 1 will probably not refer to this book often myself, but I would not hesitate to recommend it to someone who just needs a feel for how statistical concepts work. If the “why’s” of statistics have never made sense to you, this book should help.

Roger M. SAUTER Boeing Commercial Airplane Group

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