This is an excellent book on exponential families. It covers not only the basic properties of exponential families but also several modern topics such as graphical models and random networks. The author blends theories and applications elegantly and provides several useful examples from various scientific domains. It is suitable for a one-semester graduate-level course and will be an excellent reference for topic courses such as stochastic modeling and parametric models.
The author’s goal is to provide a comprehensive introduction to exponential families and related research areas. Instead of digging into technical details, the author covers the essential theories and provides examples from various applied disciplines. The author designs examples and exercises in an elegant way such that they are often motivated by practical problems but can be computed without introducing a real dataset.
This book can be roughly divided into two parts: the core concepts of exponential families and the applications of exponential families. Chapters 1–6 are the core concepts of exponential families. The author covers the statistical procedures and properties of exponential families. In Chapters 7–14, the author discusses special topics of exponential families, including missing/incomplete data, graphical models, network models, item response theory, and point processes. All these special topics are popular research areas in the modern statistical community.
This book is featured with many concepts on statistical thinking. The author often starts a section/topic with some comments on the limitations of the previous sections/topics and then smoothly introduces a new concept/method to resolve the issue. It is easy to follow the reasoning that the author made to understand why we need a generalization or a new concept. A bonus of this book is appendix A, where the author reviews key concepts and principles of statistics.
This book is well-written. The author keeps only the essential theories and augments the book with several amazing examples. Often a theoretical augment will be followed by a several examples highlighting the value of the theory. The examples in this book are not limited to a particular scientific domain; they are often from a wide variety of areas highlighting the wide-range applicability of the exponential families. The exercises are also carefully crafted; many of them connect concepts from different domains.
I found Chapters 10 and 11 to be particularly interesting. They focus on graphical models and network models; both are very popular research areas nowadays. However, there are only a few textbooks in statistics covering both topics. In Chapter 10, the author discusses two important models: the Gaussian graphical model and the log-linear model. Both models belong to exponential families, and conditional independence can be expressed as setting certain parameters to be zero, which shows a beautiful connection back to Chapter 5. Chapter 11 considers the statistical network model, which is among the fastest-growing topics in modern research. Starting with the famous Erdős–Rényi graph, the author provides an elegant introduction of the ERGM (exponential random graph models), a popular statistical model on network sciences. I would recommend anyone interested in learning graphical models or network models to read these chapters.
I would recommend this book to diverse audiences. For first-year graduate students in statistics or students from other disciplines, I would recommend reading this book chapter by chapter to learn a full picture of the exponential families and the underlying philosophy. For researchers or senior students who are familiar with exponential families, I would recommend Chapters 7–14. These chapters cover the intersection of cutting-edge research areas in statistics and exponential families.
University of Washington