**I** received (in the same box) two mathematical statistics books from CRC Press, *Understanding Advanced Statistical Methods* by Westfall and Henning, and *Statistical Theory A Concise Introduction* by Abramovich and Ritov. For review in CHANCE. While they are both decent books for teaching mathematical statistics at undergraduate borderline graduate level, I do not find enough of a novelty in them to proceed to a full review. (Given more time, I could have changed my mind about the first one.) Instead, I concentrate here on their processing of the Bayesian paradigm, which takes a wee bit more than a chapter in either of them. (And this can be done over a single métro trip!) The important following disclaimer applies: *comparing both books is highly unfair in that it is only because I received them together. They do not necessarily aim at the same audience. And I did not read the whole of either of them.*

**F**irst, the concise *Statistical Theory* covers the topic in a fairly traditional way. It starts with a warning about the philosophical nature of priors and posteriors, which reflect beliefs rather than frequency limits (just like likelihoods, no?!). It then introduces priors with the criticism that priors are difficult to build and assess. The two classes of priors analysed in this chapter are unsurprisingly conjugate priors (which hyperparameters have to be determined or chosen or estimated in the empirical Bayes heresy *[my words!, not the authors’]*) and “noninformative (objective) priors”. The criticism of the flat priors is also traditional and leads to the group invariant (Haar) measures, then to Jeffreys non-informative priors (with the apparent belief that Jeffreys only handled the univariate case). Point estimation is reduced to posterior expectations, confidence intervals to HPD regions, and testing to posterior probability ratios (with a warning about improper priors). Bayes rules make a reappearance in the following decision-theory chapter, as providers of both admissible and minimax estimators. This is it, as Bayesian techniques are not mentioned in the final “Linear Models” chapter. As a newcomer to statistics, I think I would be as bemused about Bayesian statistics as when I got my 15mn entry as a student, because here was a method that seemed to have a load of history, an inner coherence, and it was mentioned as an oddity in an otherwise purely non-Bayesian course. What good could this do to the understanding of the students?! So I would advise against getting this “token Bayesian” chapter in the book…

“You are not ignorant! Prior information is what you know prior to collecting the data.”

Understanding Advanced Statistical Methods(p.345)

**S**econd, *Understanding Advanced Statistical Methods* offers a more intuitive entry, by justifying prior distributions as summaries of prior information. And observations as a mean to increase your knowledge about the parameter. The Bayesian chapter uses a toy but very clear survey examplew to illustrate the passage from prior to posterior distributions. And to discuss the distinction between informative and noninformative priors. (I like the “Ugly Rule of Thumb” insert, as it gives a guideline without getting too comfy about it… E.g., using a 90% credible interval is good enough on p.354.) Conjugate priors are mentioned as a result of past computational limitations and simulation is hailed as a highly natural tool for analysing posterior distributions. Yay! A small section discusses the purpose of vague priors without getting much into details and suggests to avoid improper priors by using “distributions with extremely large variance”, a concept we dismissed in Bayesian Core! For how large is “extremely large”?!

“You may end up being surprised to learn in later chapters (..) that, with classical methods, you simply cannot perform the types of analyses shown in this section (…) And that’s the answer to the question, “What good is Bayes?””

Understanding Advanced Statistical Methods(p.345)

**T**hen comes the really appreciable part, a section entitled “What good is Bayes?”—it actually reads “What Good is Bayes?” (p.359), leading to a private if grammatically poor joke since I.J. Good was one of the first modern Bayesians, working with Turing at Bletchley Park…— The authors simply skip the philosophical arguments to give the reader a showcase of examples where the wealth of the Bayesian toolbox: logistic regression, VaR (Value at Risk), stock prices, drug profit prediction. Concluding with arguments in favour of the frequentist methods: (a) not requiring priors, (b) easier with generic distributifrequentistons, (c) easier to understand with simulation, and (d) easier to validate with validation. I do not mean to get into a debate about those points as my own point is that the authors are taking a certain stand about the pros and cons of the frequentist/Bayesian approaches and that they are making their readers aware of it. (Note that the Bayesian chapter comes *before* the frequentist chapter!) A further section is “Comparing the Bayesian and frequentist paradigms?” (p.384), again with a certain frequentist slant, but again making the distinctions and similarities quite clear to the reader. Of course, there is very little (if any) about Bayesian approaches in the next chapters but this is somehow coherent with the authors’ perspective. Once more, a perspective that is well-spelled and comprehensible for the reader. Even the novice statistician. In that sense, having a Bayesian chapter inside a general theory book makes sense. (The second book has a rather detailed website, by the way! Even though handling simulations in Excel and drawing graphs in SAS could be dangerous to your health…)