Why does anyone need another book in Bayesian statistics? It seems that there already exist a lot of resources for those interested in the topic. There are many excellent books covering specific aspects of Bayesian analysis or providing a wide and comprehensive background of the entire field: Berger, Bernardo and Smith, Gamerman and Freitas Lopes, Gelman et al., Robert and Cassella, and many others. Most of these books, though, will assume a certain mathematical and statistical background and would rather fit a reader’s profile of a graduate or advanced graduate level. Out of those aimed at a less sophisticated audiences, we would certainly recommend excellent books of William Bolstad, John Kruschke, and Peter Lee. There also exist some very good books on copulas: comprehensive coverage by Nelsen and Joe, and also more application-related Cherubini et al., Emrechts et al., and some others. However, instead of just referring to these works and returning to our extensive to-do lists, we decided to spend considerable amount of time and effort putting together another book—the book we presently offer to the reader.

The main reason for our endeavor is: we target a very specific audience, which as we believe is not sufficiently serviced yet with Bayesian literature. We communicate with members of this audience routinely in our day-to-day work, and we have not failed to register that just providing them with reading recommendations does not seem to satisfy their needs. Our perceived audience could be loosely divided into two groups. The first includes advanced undergraduate students of Statistics, who in all likelihood have already had some exposure to main probabilistic and statistical principles and concepts (most likely, in classical or “frequentist” setup), and may (as we probably all do) exhibit some traces of Bayesian philosophy as applicable to their everyday lives. But for them these two: statistical methods on one hand and Bayesian thinking on the other, belong to very different spheres and do not easily combine in their decision-making process.

The second group consists of practitioners of statistical methods, working in their everyday lives on specific problems requiring the use of advanced quantitative analysis. They may be aware of a Bayesian alternative to classical methods and find it vaguely attractive, but are not familiar enough with formal Bayesian analysis in order to put it to work. These practitioners populate analytical departments of banks, insurance companies, and other major businesses. In short, they might be involved in predictive modeling, quantitative forecasting, and statistical reporting which often directly call for Bayesian approach.

In the recent years, we have frequently encountered representatives of both groups described above as our collaborators, be it in undergraduate research or in applied consulting projects, or both at once (such things do happen). We have discovered a frequent need to provide to them a crash course in Bayesian methods: prior and posterior, Bayes estimation, prediction, MCMC, Bayesian regression and time series, Bayesian analysis of statistical dependence. From this environment we get the idea to concisely summarize the methodology we normally share with our collaborators in order to provide the framework for successful joint projects. Later on this idea transformed itself into a phantasy to write this book and hand it to these two segments of the audience as a potentially useful resource. This intention determines the content of the book and dictates the necessity to cover specific topics in specific order, trying also to avoid any unnecessary detail. That is why we do not include a serious introduction to probability and classical statistics (we believe that our audience has at least some formal knowledge of the main principles and facts in these fields). Instead, in Chapter 1 we just offer a review of the concepts we will eventually use. If this review happens to be insufficient to some readers, it will hopefully at least inspire them to hit the introductory books which will provide a more comprehensive coverage.

Chapter 2 deals with the basics of Bayesian statistics: prior information and experimental data, prior and posterior distributions, with emphasis on Bayesian parametric estimation, just barely touching Bayesian hypothesis testing. Some time is spend on addressing subjective versus objective Bayesian paradigms and brief discussion of noninformative priors. We spend just so much time with conjugate priors and analytical derivation of Bayes estimators that will give an idea of the scope and limitations of the analytical approach. It seems likely that most readers in their practical applications will require the use of MCMC—Markov chain Monte Carlo method—the most efficient tool in the hands of modern Bayesians. Therefore, Chapter 3 contains the basic mathematical background on both Markov chains and Monte Carlo integration and simulation. In our opinion, successful use of Markov chain Monte Carlo methods is heavily based on good understanding on these two components. Speaking of Monte Carlo methods, the central idea of variance reduction nicely transitions us to MCMC and its diagnostics. Equally important, both Markov chains and Monte Carlo methods have numerous important applications outside of Bayesian setting, and these applications will be discussed as examples.

Chapter 4 covers MCMC per se. It may look suspicious from traditional point of view that we do not particularly emphasize Gibbs sampling, rather deciding to dwell on Metropolis–Hastings algorithm in its two basic versions: independent Metropolis and random walk Metropolis-Hastings. In our opinion, this approach allows us to minimize the theoretical exposure and get close to the point using simple examples. Also, in more advanced examples at the end of the book, Gibbs sampling will rarely work without Metropolis. Another disclosure we have to make: there exists a huge library of MCMC computer programs, including those available online as freeware. All necessary references are given in the text, including OpenBUGs and several R packages. However, we also introduce some very rudimentary computer code which allows the readers to “get inside” the algorithms. This expresses the authors’ firm belief that do-it-yourself is often the best way if not to actually apply statistical computing, but at least to learn how to use it.

This might be a good time to explain the authors’ attitude to the use of computer software while reading the book. Clearly, working on the exercises, many readers would find it handy to use some computing tools, and many readers would like to use the software of their choice (be it SPSS, Matlab, Stata, or any other packages). We try to structure our exercises and text examples in a way that makes it as easy as possible. What we offer from our perspective, in addition to this possibility, is a number of illustrations containing code and/or outputs in Microsoft Excel, Mathematica, and R. Our choice of Mathematica for providing graphics is simply explained by the background of the audience of the short courses where the material of the book has been approbated. We found it hard to refuse to treat ourselves and the readers to nice Mathematica graphical tools. R might be the software of choice for modern statisticians. Therefore we find its use along with this book both handy and inevitable. We can take only limited responsibility for the readers’ introduction to R, restricting ourselves to specific uses of this language to accompany the book. There exist a number of excellent R tutorials, both in print and online, and all necessary references are provided. We can especially recommend the R book by Crawley, and a book by Robert and Casella.

The first four chapters form the first part of the book which can be used as a general introduction to Bayesian statistics with a clear emphasis on the parametric estimation. Now we need to explain what is included in the remaining chapters, and what the link is between the book’s two parts. Our world is a complicated one. Due to the recent progress of communication tools and the globalization of the world economy and information space, we humans are less and less like separate universes leading our independent lives (was it ever entirely true?) Our physical lives, our economical existences, our information fields become more and more interrelated. Many processes successfully modeled in the past by probability and statistical methods assuming independent behavior of the components, become more and more intertwined. Brownian motion is still around, as well as Newtonian mechanics is, but it often fails to serve as a good model for many complicated systems with component interactions. This explains an increased interest to model statistical dependence: be it dependence of physical lives in demography and biology, dependence of financial markets, or dependence between the components of a complex engineering system. Out of the many models of statistical dependence, copulas play a special role. They provide an attractive alternative to such traditional tools as correlation analysis or Cox’s proportional hazards. The key factor in the popularity of copulas in applications to risk management is the way they model entire joint distribution function and are not limited to its moments. This allows for the treatment of nonlinear dependence including joint tail dependence and going far beyond the standard analysis of correlation. The limitations of more traditional correlation-based approaches to modeling risks were felt around the world during the last financial crisis.

Chapter 5 is dedicated to the brief survey of pre-copula dependence models, providing necessary background for Chapter 6, where the main definitions and notations of copula models are summarized. Special attention is dedicated to a somewhat controversial problem of model selection. Here, due to a wide variety of points of view expressed in the modern literature, the authors have to narrow down the survey in sync with their (maybe subjective) preferences.

Two types of copulas most popular in applications: Gaussian copulas and copulas from Archimedean family (Clayton, Frank, Gumbel–Hougaard, and some others) are introduced and compared from model selection standpoint in Chapter 7. Suggested principles of model selection have to be illustrated by more than just short examples. This explains the emergence of the last two sections of Chapters 7 and 8, which contain some cases dealing with particular risk management problems. The choice of the cases has to do with the authors’ recent research and consulting experience. The purpose of this chapter is to provide the readers with an opportunity to follow the procedures of multivariate data analysis and copula modeling step-by-step enabling them to use these cases as either templates or insights for their own applied research studies. The authors do not take on the ambitious goal to review the state-of-the-art Bayesian statistics or copula models of dependence. The emphasis is clearly made on applications of Bayesian analysis to copula modeling, which are still remarkably rare due to the factors discussed above as well as possibly to some other reasons unknown to the authors. The main focus of the book is on equipping the readers with the tools allowing them to implement the procedures of Bayesian estimation in copula models of dependence. These procedures seem to provide a path (one of many) into the statistics of the near future. The omens which are hard to miss: copulas found their way into Basel Accord II documents regulating the world banking system, Bayesian methods are mentioned in recent FDA recommendations.

The material of the book was presented in various combinations as the content of special topic courses at the both schools where the authors teach: the University of St. Thomas in Minnesota, USA, and Astrakhan State University in Astrakhan, Russia. Additionally, parts of this material have been presented as topic modules in graduate programs at MCFAM—the Minnesota Center for Financial and Actuarial Mathematics at the University of Minnesota and short courses at the Research University High School of Economics in Moscow, Russia, and U.S. Bank in Minneapolis, MN.

We can recommend this book as a text for a full one-semester course for advanced undergraduates with some background in probability and statistics. Part I (Chapters 1–4) can be used separately as an introduction to Bayesian statistics, while Part II (Chapters 5–8) can be used separately as an introduction to copula modeling for students with some prior knowledge of Bayesian statistics. We can also suggest it to accompany topics courses for students in a wide range of graduate programs. Each chapter is equipped with a separate reference list and Chapters 2–8 by their own sets of end-of-the-chapter exercises. The companion website contains Appendices: data files and demo files in Microsoft Excel, some simple code in R, and selected exercise solutions.