**J**ust to mention the latest issue of CHANCE dedicated to the statistical issues related with slavery, edited in collaboration with the Walk Free Foundation. (I remember discussing the possibility of such an issue at the CHANCE editors meeting at JSM, Boston. I also remember Bernard Silverman discussing the case as Senior Scientist to the UK Government.) Difficulties range from defining slavery, to estimating the number of slaves, for instance by capture-mark-recapture methods. to designing ways to protect against slavery. (A stunning figure is the estimated 180,000 slaves in Poland and 20,000 in The Netherlands…)

## Archive for CHANCE

## CHANCE on modern slavery

Posted in Books, Kids, Statistics with tags CHANCE, slavery, the Netherlands, University of Warwick, Walk Free Foundation on December 23, 2017 by xi'an## Practicals of Uncertainty [book review]

Posted in Books, Statistics, University life with tags applied Bayesian analysis, Bayes factor, Bayesian foundations, book review, CHANCE, Jay Kadane, principles of uncertainty, subjective versus objective Bayes on December 22, 2017 by xi'an**O**n my way to the O’Bayes 2017 conference in Austin, I [paradoxically!] went through Jay Kadane’s Pragmatics of Uncertainty, which had been published earlier this year by CRC Press. The book is to be seen as a practical illustration of the Principles of Uncertainty Jay wrote in 2011 (and I reviewed for CHANCE). The avowed purpose is to allow the reader to check through Jay’s applied work whether or not he had “made good” on setting out clearly the motivations for his subjective Bayesian modelling. (While I presume the use of the same P of U in both books is mostly a coincidence, I started wondering how a third P of U volume could be called. *Perils of Uncertainty*? *Peddlers of Uncertainty*? The game is afoot!)

The structure of the book is a collection of fifteen case studies undertaken by Jay over the past 30 years, covering paleontology, survey sampling, legal expertises, physics, climate, and even medieval Norwegian history. Each chapter starts with a short introduction that often explains how he came by the problem (most often as an interesting PhD student consulting project at CMU), what were the difficulties in the analysis, and what became of his co-authors. As noted by the author, the main bulk of each chapter is the reprint (in a unified style) of the paper and most of these papers are actually and freely available on-line. The chapter always concludes with an epilogue (or post-mortem) that re-considers (very briefly) what had been done and what could have been done and whether or not the Bayesian perspective was useful for the problem (unsurprisingly so for the majority of the chapters!). There are also reading suggestions in the other P of U and a few exercises.

“The purpose of the book is philosophical, to address, with specific examples, the question of whether Bayesian statistics is ready for prime time. Can it be used in a variety of applied settings to address real applied problems?”

The book thus comes as a logical complement of the Principles, to demonstrate how Jay himself did apply his Bayesian principles to specific cases and how one can set the construction of a prior, of a loss function or of a statistical model in identifiable parts that can then be criticised or reanalysed. I find browsing through this series of fourteen different problems fascinating and exhilarating, while I admire the dedication of Jay to every case he presents in the book. I also feel that this comes as a perfect complement to the earlier P of U, in that it makes refering to a complete application of a given principle most straightforward, the problem being entirely described, analysed, and in most cases solved within a given chapter. A few chapters have discussions, being published in the Valencia meeting proceedings or another journal with discussions.

While all papers have been reset in the book style, I wish the graphs had been edited as well as they do not always look pretty. Although this would have implied a massive effort, it would have also been great had each chapter and problem been re-analysed or at least discussed by another fellow (?!) Bayesian in order to illustrate the impact of individual modelling sensibilities. This may however be a future project for a graduate class. Assuming all datasets are available, which is unclear from the text.

“We think however that Bayes factors are overemphasized. In the very special case in which there are only two possible “states of the world”, Bayes factors are sufficient. However in the typical case in which there are many possible states of the world, Bayes factors are sufficient only when the decision-maker’s loss has only two values.” (p. 278)

The above is in Jay’s reply to a comment from John Skilling regretting the absence of marginal likelihoods in the chapter. Reply to which I completely subscribe.

*[Usual warning: this review should find its way into CHANCE book reviews at some point, with a fairly similar content.]*

## 10 great ideas about chance [book preview]

Posted in Books, pictures, Statistics, University life with tags Abraham Wald, Alan Turing, Allais' paradox, Alonzo Church, Andrei Kolmogorov, BFF4, book review, Borel-Kolmogorov paradox, Brian Skyrms, Bruno de Finetti, Cardano's formula, CHANCE, David Hume, Dutch book argument, equiprobability, exchangeability, Frank Ramsey, gambling, Gerolamo Cardano, Henri Poincaré, heuristics, Jakob Bernoulli, John Maynard Keynes, John von Neumann, Karl Popper, Martin-Löf, measure theory, p-values, Persi Diaconis, Pierre Simon Laplace, PUP, Radon-Nikodym Theorem, randomness, Richard von Mises, sufficiency, Thomas Bayes, Venn diagram on November 13, 2017 by xi'an*[As I happened to be a reviewer of this book by Persi Diaconis and Brian Skyrms, I had the opportunity (and privilege!) to go through its earlier version. Here are the [edited] comments I sent back to PUP and the authors about this earlier version. All in all, a terrific book!!!]*

**T**he historical introduction (“measurement”) of this book is most interesting, especially its analogy of chance with length. I would have appreciated a connection earlier than Cardano, like some of the Greek philosophers even though I gladly discovered there that Cardano was not only responsible for the closed form solutions to the third degree equation. I would also have liked to see more comments on the vexing issue of *equiprobability*: we all spend (if not waste) hours in the classroom explaining to (or arguing with) students why their solution is not correct. And they sometimes never get it! [And we sometimes get it wrong as well..!] Why is such a simple concept so hard to explicit? In short, but this is nothing but a personal choice, I would have made the chapter more conceptual and less chronologically historical.

“Coherence is again a question of consistent evaluations of a betting arrangement that can be implemented in alternative ways.” (p.46)

The second chapter, about Frank Ramsey, is interesting, if only because it puts this “man of genius” back under the spotlight when he has all but been forgotten. (At least in my circles.) And for joining probability and utility together. And for postulating that probability can be derived from expectations rather than the opposite. Even though betting or gambling has a (negative) stigma in many cultures. At least gambling for money, since most of our actions involve some degree of betting. But not in a rational or reasoned manner. (Of course, this is not a mathematical but rather a psychological objection.) Further, the justification through betting is somewhat tautological in that it assumes probabilities are true probabilities from the start. For instance, the Dutch book example on p.39 produces a gain of .2 only if the probabilities are correct.

> gain=rep(0,1e4) > for (t in 1:1e4){ + p=rexp(3);p=p/sum(p) + gain[t]=(p[1]*(1-.6)+p[2]*(1-.2)+p[3]*(.9-1))/sum(p)} > hist(gain)

As I made it clear at the BFF4 conference last Spring, I now realise I have never really adhered to the Dutch book argument. This may be why I find the chapter somewhat unbalanced with not enough written on utilities and too much on Dutch books.

“The force of accumulating evidence made it less and less plausible to hold that subjective probability is, in general, approximate psychology.” (p.55)

A chapter on “psychology” may come as a surprise, but I feel *a posteriori* that it is appropriate. Most of it is about the Allais paradox. Plus entries on Ellesberg’s distinction between risk and uncertainty, with only the former being quantifiable by “objective” probabilities. And on Tversky’s and Kahneman’s distinction between heuristics, and the framing effect, i.e., how the way propositions are expressed impacts the choice of decision makers. However, it is leaving me unclear about the conclusion that the fact that people behave irrationally should not prevent a reliance on utility theory. Unclear because when taking actions involving other actors their potentially irrational choices should also be taken into account. (This is mostly nitpicking.)

“This is Bernoulli’s swindle. Try to make it precise and it falls apart. The conditional probabilities go in different directions, the desired intervals are of different quantities, and the desired probabilities are different probabilities.” (p.66)

The next chapter (“frequency”) is about Bernoulli’s Law of Large numbers and the stabilisation of frequencies, with von Mises making it the basis of his approach to probability. And Birkhoff’s extension which is capital for the development of stochastic processes. And later for MCMC. I like the notions of “disreputable twin” (p.63) and “Bernoulli’s swindle” about the idea that “chance is frequency”. The authors call the identification of probabilities as limits of frequencies Bernoulli‘s swindle, because it cannot handle zero probability events. With a nice link with the testing fallacy of equating rejection of the null with acceptance of the alternative. And an interesting description as to how Venn perceived the fallacy but could not overcome it: “If Venn’s theory appears to be full of holes, it is to his credit that he saw them himself.” The description of von Mises’ Kollectiven [and the welcome intervention of Abraham Wald] clarifies my previous and partial understanding of the notion, although I am unsure it is that clear for all potential readers. I also appreciate the connection with the very notion of *randomness* which has not yet found I fear a satisfactory definition. This chapter asks more (interesting) questions than it brings answers (to those or others). But enough, this is a brilliant chapter!

“…a random variable, the notion that Kac found mysterious in early expositions of probability theory.” (p.87)

Chapter 5 (“mathematics”) is very important [from my perspective] in that it justifies the necessity to associate measure theory with probability if one wishes to evolve further than urns and dices. To entitle Kolmogorov to posit his axioms of probability. And to define properly conditional probabilities as random variables (as my third students fail to realise). I enjoyed very much reading this chapter, but it may prove difficult to read for readers with no or little background in measure (although some advanced mathematical details have vanished from the published version). Still, this chapter constitutes a strong argument for preserving measure theory courses in graduate programs. As an aside, I find it amazing that mathematicians (even Kac!) had not at first realised the connection between measure theory and probability (p.84), but maybe not so amazing given the difficulty many still have with the notion of conditional probability. (Now, I would have liked to see some description of Borel’s paradox when it is mentioned (p.89).

“Nothing hangs on a flat prior (…) Nothing hangs on a unique quantification of ignorance.” (p.115)

The following chapter (“inverse inference”) is about Thomas Bayes and his posthumous theorem, with an introduction setting the theorem at the centre of the Hume-Price-Bayes triangle. (It is nice that the authors include a picture of the original version of the essay, as the initial title is much more explicit than the published version!) A short coverage, in tune with the fact that Bayes only contributed a twenty-plus paper to the field. And to be logically followed by a second part [formerly another chapter] on Pierre-Simon Laplace, both parts focussing on the selection of prior distributions on the probability of a Binomial (coin tossing) distribution. Emerging into a discussion of the position of statistics within or even outside mathematics. (And the assertion that Fisher was the Einstein of Statistics on p.120 may be disputed by many readers!)

“So it is perfectly legitimate to use Bayes’ mathematics even if we believe that chance does not exist.” (p.124)

The seventh chapter is about Bruno de Finetti with his astounding representation of exchangeable sequences as being mixtures of iid sequences. Defining an implicit prior on the side. While the description sticks to binary events, it gets quickly more advanced with the notion of partial and Markov exchangeability. With the most interesting connection between those exchangeabilities and sufficiency. (I would however disagree with the statement that “Bayes was the father of parametric Bayesian analysis” [p.133] as this is extrapolating too much from the Essay.) My next remark may be non-sensical, but I would have welcomed an entry at the end of the chapter on cases where the exchangeability representation fails, for instance those cases when there is no sufficiency structure to exploit in the model. A bonus to the chapter is a description of Birkhoff’s ergodic theorem “as a generalisation of de Finetti” (p..134-136), plus half a dozen pages of appendices on more technical aspects of de Finetti’s theorem.

“We want random sequences to pass all tests of randomness, with tests being computationally implemented”. (p.151)

The eighth chapter (“algorithmic randomness”) comes (again!) as a surprise as it centres on the character of Per Martin-Löf who is little known in statistics circles. (The chapter starts with a picture of him with the iconic Oberwolfach sculpture in the background.) Martin-Löf’s work concentrates on the notion of randomness, in a mathematical rather than probabilistic sense, and on the algorithmic consequences. I like very much the section on random generators. Including a mention of our old friend RANDU, the 16 planes random generator! This chapter connects with Chapter 4 since von Mises also attempted to define a random sequence. To the point it feels slightly repetitive (for instance Jean Ville is mentioned in rather similar terms in both chapters). Martin-Löf’s central notion is computability, which forces us to visit Turing’s machine. And its role in the undecidability of some logical statements. And Church’s recursive functions. (With a link not exploited here to the notion of probabilistic programming, where one language is actually named Church, after Alonzo Church.) Back to Martin-Löf, (I do not see how his test for randomness can be implemented on a real machine as the whole test requires going through the entire sequence: since this notion connects with von Mises’ Kollektivs, I am missing the point!) And then Kolmororov is brought back with his own notion of complexity (which is also Chaitin’s and Solomonov’s). Overall this is a pretty hard chapter both because of the notions it introduces and because I do not feel it is completely conclusive about the notion(s) of randomness. A side remark about casino hustlers and their “exploitation” of weak random generators: I believe Jeff Rosenthal has a similar if maybe simpler story in his book about Canadian lotteries.

“Does quantum mechanics need a different notion of probability? We think not.” (p.180)

The penultimate chapter is about Boltzmann and the notion of “physical chance”. Or statistical physics. A story that involves Zermelo and Poincaré, And Gibbs, Maxwell and the Ehrenfests. The discussion focus on the definition of probability in a thermodynamic setting, opposing time frequencies to space frequencies. Which requires ergodicity and hence Birkhoff [no surprise, this is about ergodicity!] as well as von Neumann. This reaches a point where conjectures in the theory are yet open. What I always (if presumably naïvely) find fascinating in this topic is the fact that ergodicity operates without requiring randomness. Dynamical systems can enjoy ergodic theorem, while being completely deterministic.) This chapter also discusses quantum mechanics, which main tenet requires probability. Which needs to be defined, from a frequency or a subjective perspective. And the Bernoulli shift that brings us back to random generators. The authors briefly mention the Einstein-Podolsky-Rosen paradox, which sounds more metaphysical than mathematical in my opinion, although they get to great details to explain Bell’s conclusion that quantum theory leads to a mathematical impossibility (but they lost me along the way). Except that we “are left with quantum probabilities” (p.183). And the chapter leaves me still uncertain as to why statistical mechanics carries the label *statistical*. As it does not seem to involve inference at all.

“If you don’t like calling these ignorance priors on the ground that they may be sharply peaked, call them nondogmatic priors or skeptical priors, because these priors are quite in the spirit of ancient skepticism.” (p.199)

And then the last chapter (“induction”) brings us back to Hume and the 18th Century, where somehow “everything” [including statistics] started! Except that Hume’s strong scepticism (or skepticism) makes induction seemingly impossible. (A perspective with which I agree to some extent, if not to Keynes’ extreme version, when considering for instance financial time series as stationary. And a reason why I do not see the criticisms contained in the Black Swan as pertinent because they savage normality while accepting stationarity.) The chapter rediscusses Bayes’ and Laplace’s contributions to inference as well, challenging Hume’s conclusion of the impossibility to finer. Even though the representation of ignorance is not unique (p.199). And the authors call again for de Finetti’s representation theorem as bypassing the issue of whether or not there is such a thing as chance. And escaping inductive scepticism. (The section about Goodman’s grue hypothesis is somewhat distracting, maybe because I have always found it quite artificial and based on a linguistic pun rather than a logical contradiction.) The part about (Richard) Jeffrey is quite new to me but ends up quite abruptly! Similarly about Popper and his exclusion of induction. From this chapter, I appreciated very much the section on skeptical priors and its analysis from a meta-probabilist perspective.

There is no conclusion to the book, but to end up with a chapter on induction seems quite appropriate. (But there is an appendix as a probability tutorial, mentioning Monte Carlo resolutions. Plus notes on all chapters. And a commented bibliography.) Definitely recommended!

*[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]*

## errors, blunders, and lies [book review]

Posted in Books, Kids, Statistics, University life with tags and lies, ASA, blunders, book review, CHANCE, CRC Press, errors, introductory textbooks, Pierre Simon Laplace, The Lady Tasting Tea on July 9, 2017 by xi'an **T**his new book by David Salsburg is the first one in the ASA-CRC Series on Statistical Reasoning in Science and Society. Which explains why I heard about it both from CRC Press [as a suggested material for a review in CHANCE] and from the ASA [as mass emailing]. The name of the author did not ring a bell until I saw the line about his earlier The Lady Tasting Tea book, a best-seller in the category of “soft [meaning math- and formula-free] introduction to Statistics through picturesque characters”. Which I did not read either [but Bob Carpenter did].

The current book is of the same flavour, albeit with some maths formulas [each preceded by a lengthy apology for using maths and symbols]. The topic is the one advertised in the title, covering statistical errors and the way to take advantage of them, model mis-specification and robustness, and the detection of biases and data massaging. I read the short book in one quick go, waiting for the results of the French Legislative elections, and found no particular appeal in the litany of examples, historical entries, pitfalls, and models I feel I have already read so many times in the story-telling approach to statistics. (Naked Statistics comes to mind.)

It is not that there anything terrible with the book, which is partly based on the author’s own experience in a pharmaceutical company, but it does not seem to bring out any novelty for engaging into the study of statistics or for handling data in a more rational fashion. And I do not see which portion of the readership is targeted by the book, which is too allusive for academics and too academic for a general audience, who is not necessarily fascinated by the finer details of the history (and stories) of the field. As in The Lady Tasting Tea, the chapters constitute a collection of vignettes, rather than a coherent discourse leading to a theory or defending an overall argument. Some chapters are rather poor, like the initial chapter explaining the distinction between lies, blunders, and errors through the story of the measure of the distance from Earth to Sun by observing the transit of Venus, not that the story is uninteresting, far from it!, but I find it lacking in connecting with statistics [e.g., the meaning of a “correct” observation is never explained]. Or the chapter on the Princeton robustness study, where little is explained about the nature of the wrong distributions, which end up as specific contaminations impacting mostly the variance. And some examples are hardly convincing, like those on text analysis (Chapters 13, 14, 15), where there is little backup for using Benford’s law on such short datasets. Big data is understood only under the focus of large p, small n, which is small data in my opinion! (Not to mention a minor crime de *lèse-majesté* in calling Pierre-Simon Laplace Simon-Pierre Laplace! I would also have left the *Marquis de* aside as this title came to him during the Bourbon Restauration, despite him having served Napoléon for his entire reign.) And, as mentioned above, the book contains apologetic mathematics, which never cease to annoy me since apologies are not needed. While the maths formulas are needed.

## lost in translation [oops]

Posted in Statistics with tags book review, CHANCE, Maurice Audin, Michèle Audin, One Hundred Twenty-One Days, translation, Une Vie Brève on April 20, 2017 by xi'an**I**n my latest book review for CHANCE, I published my review of Michèle Audin’s Une Vie Brève. However, for a reason I cannot remember, I (?) added that the translated edition of the book was One Hundred Twenty-One Days, which is an altogether different book by the same author! (I actually cannot find a trace of this reference in my submitted LaTeX file, although this does not signify I did not add the remark when I got the proofs.) This was pointed out by the translator, Christiana Hills, so apologies to the author and to the readers for this confusion!

## truth or truthiness [book review]

Posted in Books, Kids, pictures, Statistics, University life with tags Andrew Gelman, Cambridge University Press, causality, CHANCE, data science, Don Rubin, fracking, Howard Wainer, Oklahoma, testing, tribune, truthiness, University of Warwick on March 21, 2017 by xi'an**T**his 2016 book by Howard Wainer has been sitting (!) on my desk for quite a while and it took a long visit to Warwick to find a free spot to quickly read it and write my impressions. The subtitle is, as shown on the picture, *“Distinguishing fact from fiction by learning to think like a data scientist”*. With all due respect to the book, which illustrates quite pleasantly the dangers of (pseudo-)data mis- or over- (or eve under-)interpretation, and to the author, who has repeatedly emphasised those points in his books and ~~tribunes~~ opinion columns, including those in CHANCE, I do not think the book teaches how to think like a data scientist. In that an arbitrary neophyte reader would not manage to handle a realistic data centric situation without deeper training. But this collection of essays, some of which were tribunes, makes for a nice reading nonetheless.

I presume that in this post-truth and alternative facts [dark] era, the notion of *truthiness* is familiar to most readers! It is often based on a misunderstanding or a misappropriation of data leading to dubious and unfounded conclusions. The book runs through dozens of examples (some of them quite short and mostly appealing to common sense) to show how this happens and to some extent how this can be countered. If not avoided as people will always try to bend, willingly or not, the data to their conclusion.

There are several parts and several themes in Truth or Truthiness, with different degrees of depth and novelty. The more involved part is in my opinion the one about causality, with illustrations in educational testing, psychology, and medical trials. (The illustration about fracking and the resulting impact on Oklahoma earthquakes should not be in the book, except that there exist officials publicly denying the facts. The same remark applies to the testing cheat controversy, which would be laughable had not someone ended up the victim!) The section on graphical representation and data communication is less exciting, presumably because it comes *after* Tufte’s books and message. I also feel the 1854 cholera map of John Snow is somewhat over-exploited, since he only drew the map after the epidemic declined. The final chapter * Don’t Try this at Home* is quite anecdotal and at the same time this may the whole point, namely that in mundane questions thinking like a data scientist is feasible and leads to sometimes surprising conclusions!

*“In the past a theory could get by on its beauty; in the modern world, a successful theory has to work for a living.” (p.40)*

The book reads quite nicely, as a whole and a collection of pieces, from which class and talk illustrations can be borrowed. I like the “learned” tone of it, with plenty of citations and witticisms, some in Latin, Yiddish and even French. (Even though the later is somewhat inaccurate! *Si ça avait pu se produire, ça avait dû se produire* [p.152] would have sounded more vernacular in my Gallic opinion!) I thus enjoyed unreservedly Truth or Truthiness, for its rich style and critical message, all the more needed in the current times, and far from comparing it with a bag of potato chips as Andrew Gelman did, I would like to stress its classical tone, in the sense of being immersed in a broad and deep culture that seems to be receding fast.

## a concise introduction to statistical inference [book review]

Posted in Statistics with tags A concise introduction to statistical inference, Bayesian inference, book review, calculus, CHANCE, CRC Press, hypothesis testing, introductory textbooks, Jacco Thijssen, Riemann integration, subjective probability on February 16, 2017 by xi'an*[Just to warn readers and avoid emails about Xi’an plagiarising Christian!, this book was sent to me by CRC Press for a review. To be published in CHANCE.]*

**T**his is an introduction to statistical inference. And with 180 pages, it indeed is concise! I could actually stop the review at this point as a concise review of a concise introduction to statistical inference, as I do not find much originality in this introduction, intended for “mathematically sophisticated first-time student of statistics”. Although sophistication is in the eye of the sophist, of course, as this book has margin symbols in the guise of integrals to warn of section using “differential or integral calculus” and a remark that the book is still accessible without calculus… (Integral calculus as in Riemann integrals, not Lebesgue integrals, mind you!) It even includes appendices with the Greek alphabet, summation notations, and exponential/logarithms.

“In statistics we often bypass the probability model altogether and simply specify the random variable directly. In fact, there is a result (that we won’t cover in detail) that tells us that, for any random variable, we can find an appropriate probability model.” (p.17)

Given its limited mathematical requirements, the book does not get very far in the probabilistic background of statistics methods, which makes the corresponding chapter not particularly helpful as opposed to a prerequisite on probability basics. Since not much can be proven without “all that complicated stuff about for any ε>0” (p.29). And makes defining correctly notions like the Central Limit Theorem impossible. For instance, Chebychev’s inequality comes within a list of admitted results. There is no major mistake in the chapter, even though mentioning that two correlated Normal variables are jointly Normal (p.27) is inexact.

“The power of a test is the probability that you do not reject a null that is in fact correct.” (p.120)

Most of the book follows the same pattern as other textbooks at that level, covering inference on a mean and a probability, confidence intervals, hypothesis testing, p-values, and linear regression. With some words of caution about the interpretation of p-values. (And the unfortunate inversion of the interpretation of power above.) Even mentioning the Cult [of Significance] I reviewed a while ago.

Given all that, the final chapter comes as a surprise, being about Bayesian inference! Which should make me rejoice, obviously, but I remain skeptical of introducing the concept to readers with so little mathematical background. And hence a very shaky understanding of a notion like conditional distributions. (Which reminds me of repeated occurrences on X validated when newcomers hope to bypass textbooks and courses to grasp the meaning of posteriors and such. Like when asking why Bayes Theorem does not apply for expectations.) I can feel the enthusiasm of the author for this perspective and it may diffuse to some readers, but apart from being aware of the approach, I wonder how much they carry away from this brief (decent) exposure. The chapter borrows from Lee (2012, 4th edition) and from Berger (1985) for the decision-theoretic part. The limitations of the exercise are shown for hypothesis testing (or comparison) by the need to restrict the parameter space to two possible values. And for decision making. Similarly, introducing improper priors and *the likelihood principle* [distinguished there from *the law of likelihood*] is likely to get over the head of most readers and clashes with the level of the previous chapters. (And I do not think this is the most efficient way to argue in favour of a Bayesian approach to the problem of statistical inference: I have now dropped all references to the likelihood principle from my lectures. Not because of the controversy, but simply because the students do not get it.) By the end of the chapter, it is unclear a neophyte would be able to spell out how one could specify a prior for one of the problems processed in the earlier chapters. The appendix on de Finetti’s formalism on personal probabilities is very much unlikely to help in this regard. While it sounds so far beyond the level of the remainder of the book.