*Thinking Probabilistically* is a conceptual and problem-focused introduction to a wide range of topics in probability theory, and its connections with a huge range of theoretical and applied fields. The (ambitious) opening sentence sums up the philosophy of the book well: “the purposes of this book are to familiarize you with a broad range of examples where randomness plays a key role, develop an intuition for it, and get to the level where you may read a recent research paper on the subject and be able to understand the terminology, the context, and the tools used.” While it is written roughly at an introductory level for many of the topics, it assumes a reasonably sophisticated mathematical background from the intended audience – standard PDE solution methods, linear algebra, multivariable analysis, and reasonable familiarity with undergraduate-level probability. While things like random walks, the Fourier Transform, and the Central Limit Theorem are “introduced” in the text, the development is really too rapid for someone completely unfamiliar with these ideas to follow along, at least without augmenting one’s reading with a more standard and systematic text. Often these topics are introduced in the context of a problem from physics, engineering, or biology, which can be both refreshing but a bit disorienting if the reader is completely unfamiliar with the context.

Loosely, I would organize the book into three parts. Chapters 1 and 2 motivate the study of randomness and introduce important aspects of elementary probability theory such as Markov processes (and chains), random walks, and their connection to PDE. Chapters 3 through 6 then survey and connect a variety of standard topics in statistical physics and stochastic analysis from Langevin equations to extreme value statistics and rare events (i.e. long-tailed distributions), with frequent but brief discussions of applications from condensed matter physics and engineering, to cell biology and financial mathematics. Finally Chapters 7 through 9 cover topics that are more specialized than in general books covering similar topics: anomalous diffusion, a brief but surprisingly deep foray into random matrix theory, and finally a discussion of percolation theory and related tools from statistical physics. Throughout the text are copious discussions of related ideas and applications, often, though not always, with references. There are also interesting historical discussions, and several appendices which cover much of the needed mathematical background concisely.

The book takes a distinctly “applied” philosophy, explicitly stating that it will use the language and approaches common in physics, and will often eschew rigorous derivations in favor of plausible arguments. Again to quote from the introduction, “To paraphrase Feynman – in many cases, there is no need to solve a model exactly, since the connection between the model and the reality is only crude, and we made numerous (far more important) approximations in deriving the model equations (von Neumann put this more bluntly: ‘There’s no sense in being precise when you don’t even know what you’re talking about’).” This approach is, at least in my experience, relatively uncommon in probability theory or stochastic analysis. In some cases it is extremely insightful – many of the exercises require a reader to think carefully about how to define terms and formulate a model of some situation. These are excellent tools to expose students or researchers from more rigorous areas to thinking of the vague demands of real-world applications and to make an effort in transforming verbal hypotheses into operationalized mathematical models.

However, this approach is not without its pitfalls. Much of the presentation is difficult to follow without recognizing some of the ideas from elsewhere, and much of the notation is not clearly or carefully defined. For example, the introduction of Langevin equations in Chapters 3 has a “noise” term which is never really defined, and it makes use of a classical mechanics formalism in interpreting and understanding these equations (e.g. the mass “m” in equation (3.10) is never defined). Some of the scalings needed to correctly interpret stochastic differential equations are hard to motivate without having seen them before, and this repeats in later chapters with ansatzes that may be quite far from intuitive without some familiarity of the topic already. This style works well for someone who has seen these topics before and can fill in these details themselves, but it felt very difficult to follow some of the reasoning in places where my own background was not deep (such as the chapter on Random Matrix Theory). This approach made some parts of the book quite challenging, and other parts somewhat mundane. Nevertheless, a motivated reader at any level can still get some excellent insights from this text, as long as they are prepared to struggle.

Dr. Andrew Krause is an Assistant Professor in Applied and Computational Mathematics at Durham University. His research is primarily in mathematical biology and nonlinear dynamical systems. More information can be found at https://www.andrewkrause.org/.