# Wordtrade.com Mathematics

## Bayesian Method

Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition by Jeff Gill (Statistics in the Social and Behavioral Sciences: Chapman and Hall/CRC) The first edition helped pave the way for Bayesian approaches to become more prominent in social science methodology. While the focus remains on practical modeling and basic theory as well as on intuitive explanations and derivations without skipping steps, this second edition incorporates the latest methodology and recent changes in software offerings.

New to the Second Edition

• Two chapters on Markov chain Monte Carlo (MCMC) that cover ergodicity, convergence, mixing, simulated annealing, reversible jump MCMC, and coupling
• Expanded coverage of Bayesian linear and hierarchical models
• More technical and philosophical details on prior distributions
• A dedicated R package (BaM) with data and code for the examples as well as a set of functions for practical purposes such as calculating highest posterior density (HPD) intervals

Requiring only a basic working knowledge of linear algebra and calculus, this text is one of the few to offer a graduate-level introduction to Bayesian statistics for social scientists. It first introduces Bayesian statistics and inference, before moving on to assess model quality and fit. Subsequent chapters examine hierarchical models within a Bayesian context and explore MCMC techniques and other numerical methods. Concentrating on practical computing issues, the author includes specific details for Bayesian model building and testing and uses the R and BUGS software for examples and exercises.

Presents many new developments in the field, such as the deviance information criterion (DIC), hybrid MCMC algorithms, diagnostics, perfect sampling, and Bayesian nonparametrics

Emphasizes applications commonly used in the social sciences, including regression models and covariates

Explores the most up-to-date MCMC package in Win BUGS and the new MCMCpack in R

Provides detailed advice and guidance on the mechanics of stochastic simulation

Bayesian methods continue to become more important and central to sta­tistical analysis, broadly speaking. Seemingly, no issue of the Journal of the American Statistical Association arrives without at least one Bayesian application or theoretical development. While this upward trend started in the 1990s after we discovered Markov chain Monte Carlo hiding in statistical physics, the trend accelerates in the 21st Century. A nice foretelling is found in the 1999 Science article by David Malakoff, "Bayesian Boom," complete with anecdotes about popular uses in biology and computing as well as quotes from John Geweke. Back in 1995, the Bayesian luminary Bruno de Finetti predicted that by the year 2020 we would see a paradigm shift to Bayesian thinking (quoted in Smith [1995]). I believe we are fully on-track to meet this schedule.

Bayesian computing is broader and more varied than it was at the writing of the first edition. In addition to BUGS and WinBUGS, we now routinely use MCMCpack, JAGS, openbugs, bayesm, and even the new SAS MCMC procedure. The diagnostic routines in R, BOA and CODA continue to be useful and are more stable than they were. Of the course the lingua franca of R is critical, and xx, xxi many researchers use C or C++ for efficiency. Issues of statistical computing remain an important component of the book. It is also necessary to download and use the R packages CODA and BOA for MCMC diagnostics.

Bayesian approaches are also increasingly popular in related fields not di­rectly addressed in this text. There is now an interesting literature in ar­chaeology that is enjoyable to read (Reese 1994, Freeman 1976, Laxton et al. 1994), and the best starting point is the seminal paper by Litton and Buck (1995) that sets the agenda for Bayesian archaeometrics. Researchers in this area have also become frustrated with the pathetic state of the null hypothe­sis significance test in the social and behavioral sciences (Cowgill 1977). One area where Bayesian modeling is particularly useful is in archaeological foren­sics, where researchers make adult-age estimates of early humans (Lucy et al. 1996, Aykroyd et al. 1999).

Changes from the First Edition

A reader of the first edition will notice many changes in this revision. Hope­fully these constitute improvements (they certainly constituted a lot of work). First, the coverage of Markov chain Monte Carlo is greatly expanded. The rea­son for this is obvious, but bears mentioning. Modern applied Bayesian work is integrally tied to stochastic simulation and there are now several high qual­ity software alternatives for implementation. Unfortunately these solutions can be complex and the theoretical issues are often demanding. Coupling this with easy to use software, such as WinBUGS and MCMCpack, means that there are users who are unaware of the dangers inherent in MCMC work. I get a fair number of journal and book press manuscripts to review supporting this point. There is now a dedicated chapter on MCMC theory covering issues like ergodicity, convergence, and mixing. The last chapter is is an extension of sections from the first edition that now covers in greater detail tools like: sim­ulated annealing (including its many variants), reversible jump MCMC, and coupling from the past. Markov chain Monte Carlo research is an incredibly dynamic and fast growing literature and the need to get some of these ideas before a social science audience was strong. The reader will also note a sub­ stantial increase on MCMC examples and practical guidance. The objective is to provide detailed advice on day-to-day issues of implementation. Markov chain Monte Carlo is now discussed in detail in the first chapter, giving it the prominent position that it deserves. It is my belief that Gibbs sampling is as fundamental to estimation as maximum likelihood, but we (collectively) just do not realize it yet. Recall that there was about 40 years between Fisher's important papers and the publication of Birnbaum's Likelihood Principle. This second edition now provides a separate chapter on Bayesian linear mod­els. Regression remains the favorite tool of quantitative social scientists, and it makes sense to focus on the associated Bayesian issues in a full chapter. Most of the questions I get by email and at conferences are about priors, re­flecting sensitivity about how priors may affect final inferences. Hence, the chapter on forms of prior distributions is longer and more detailed. I have found that some forms are particularly well-suited to the type of work that social and behavioral researchers do. One of the strengths of Bayesian meth­ods is the ease with which hierarchical models can be specified to recognize different levels and sources in the data. So there is now an expanded chapter on this topic alone, and while Chapter 10 focuses exclusively on hierarchical model specifications, these models appear throughout the text reflecting their importance in Bayesian statistics.

Additional topics have crept into this edition, and these are covered at varied levels from a basic introduction to detailed discussions. Some of these topics are older and well-known, such as Bayesian time-series, empirical Bayes, Bayesian decision theory, additional prior specifications, model checking with posterior data prediction, the deviance information criterion (DIC), methods for computing highest posterior density (HPD) intervals, convergence theory, metropolis-coupling, tempering, reversible jump MCMC, perfect sampling, software packages related to BUGS, and additional models based on normal and Student's-t assumptions.

Some new features are more structural. There is now a dedicated R pack­age to accompany this book, BaM (for "Bayesian Methods" ). This package includes data and code for the examples as well as a set of functions for practical purposes like calculated HPD intervals. These materials and more associated with the book are available at dedicated the Washington Univer­sity website: http: //stats.wustl.edu/BMSBSA. The second edition includes three appendices covering basic maximum likelihood theory, distributions,

and BUGS software. These were moved to separate sections to make referenc­ing easier and to preserve the flow of theoretical discussions. References are now contained in a single bibliography at the end for similar reasons. Some changes are more subtle. I've changed all instances of "noninformative" to "uninformative" since the first term does not really describe prior distribu­tions. Markov chain Monte Carlo techniques are infused throughout, befitting their central role in Bayesian work. Experience has been that social science graduate students remain fairly tepid about empirical examples that focus on rats, lizards, beetles, and nuclear pumps. Furthermore, as of this writing there is no other comprehensive Bayesian text in the social sciences, outside of economics (except the out-of-print text by Phillips [1973]).

To begin, the prerequisites remain the same. Readers will need to have a basic working knowledge of linear algebra and calculus to follow many of the sections. Gill’s math text, Essential Mathematics for Political and Social Re­search (2006), provides an overview of such material. Chapter 1 gives a brief review of the probability basics required here, but it is certainly helpful to have studied this material before. Finally, one cannot understand Bayesian modeling without knowledge of maximum likelihood theory. I recognize grad­uate programs differ in their emphasis on this core material, so Appendix A covers these essential ideas.

The second edition is constructed in a somewhat different fashion than the first. The most obvious difference is that the chapter on generalized linear models has been recast as an appendix, as mentioned. Now the introduc­tory material flows directly into the construction of basic Bayesian statistical models and the procession of core ideas is not interrupted by a non-Bayesian discussion of standard models. Nonetheless, this material is important to have close at hand and hopefully the appendix approach is convenient. Another notable change is the "promotion" of linear models to their own chapter. This material is important enough to stand on its own despite the overlap with Bayesian normal and Student's-t models. Other organization changes are found in the computational section where considerable extra material has been added, both in terms of theory and practice. Markov chain Monte Carlo set the Bayesians free, and remains an extremely active research field. Keeping up with this literature is a time-consuming, but enjoyable, avocation.

There are a number of ways that a graduate course could be structured around this text. For a basic-level introductory course that emphasizes the­oretical ideas, the first seven chapters provide a detailed overview without considering many computational challenges. Some of the latter chapters are directed squarely at sophisticated social scientists who have not yet explored some of the subtle theory of Markov chains. Among the possible structures, consider the following curricula.

Basic Introductory Course

Chapter 1: Background and Introduction

Chapter 2: Specifying Bayesian Models

Chapter 3: The Normal and Student's-t Models

Chapter 4: The Bayesian Linear Model

Chapter 10: Bayesian Hierarchical Models

Thorough Course without an Emphasis on Computing

Chapter 1: Background and Introduction

Chapter 2: Specifying Bayesian Models

Chapter 3: The Normal and Student's-t Models Chapter 4: The Bayesian Linear Model

Chapter 5: The Bayesian Prior

Chapter 6: Assessing Model Quality

Chapter 7: Bayesian Hypothesis Testing and the Bayes Factor

Chapter 10: Bayesian Hierarchical Models

A Component of a Statistical Computing Course

Chapter 2: Specifying Bayesian Models

Chapter 8: Monte Carlo Methods

Chapter 9: Basics of Markov Chain Monte Carlo

Chapter 11: Some Markov Chain Monte Carlo Theory

Chapter 12: Utilitarian Markov Chain Monte Carlo

Chapter13: Advanced Markov Chain Monte Carlo

A Component of an Estimation Course:

Appendix A: Generalized Linear Model Review

Chapter I: Background and Introduction

Chapter 2: Specifying Bayesian Models

Chapter 4: The Bayesian Linear Model

Chapter 7: Bayesian Hypothesis Testing and the Bayes Factor

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