Casanova’s Lottery

Stephen Stigler

Softcover: 232 pages, 62 images, 22 tables
Publisher: University of Chicago Press (2022)
ISBN-13: 9780226820781

This “history of a revolutionary game of chance” is the latest book by Stephen Stigler and definitely of an historical nature, following the progression of the French State Lottery from its inception as the Loterie royale in 1758 to the Loterie Nationale in 1836, with the intermediate names of Loterie de France, Loterie Nationale, Loterie impériale, and Loterie royale, reflecting the agitated history of the turn of that century in France.

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The incentive for following this state lottery is that it is exceptional for its mathematical foundations. Contrary to other lotteries of the time, it was rationally grounded in averaging losses and gains in the long run (for the state). The French (Royal) state thus accepted from the start the possibility of huge losses from some draws, since these would be compensated for by even larger gains.

The reasoning proved correct: The Loterie went on to provide as much as 4% of the overall state budget, despite the costs of maintaining a network of betting places and employees, who had to be mathematically savvy to compute the exact gains of the winners and spot fraudsters. This is rather amazing because the understanding of the Law of Large Numbers was, at the time, quite fresh (from an historical perspective), thanks to the considerable advances made by Pascal, Fermat, (Jakob) Bernoulli, and a few others. (The book mentions the encyclopedist and mathematician Jean d’Alembert as being present at the meeting that decided on creation of the Loterie in 1757.)

One may wonder why Casanova gets the credit for this lottery, at least in the book title. In true agreement with (the very same) Stigler’s Law, the devising of the Loterie is directly connected with a former Genoan lottery and subsequent avatars that appeared in other Italian cities, including Casanova’s own Venezia. Jack-of-all-trades Casanova was instrumental in selling the notion to the French state, having landed in Paris after a daring flight from the Serenissima’s jails.

After succeeding in convincing the king’s officers to launch the scheme crafted by a certain Ranieri (dé) Calzabigi—not to be confused with the much-maligned Salieri—who would later collaborate with Gluck on Orfeo ed Eurydice and Alceste, Casanova received a salary from the Loterie administration. He ran several betting offices in Paris, until he left the city for further adventures—including an attempt to reproduce the lottery in Berlin, where Frederick II proved less receptive or more risk-adverse than Louis XIV (possibly due to Euler’s cautionary advice). The final sentence of the book stands by its title: “It was indeed Casanova’s lottery” (p. 210).

Unsurprisingly, given the author’s fascination with Pierre-Simon Laplace, this great man plays a role in the history, first by writing, in 1774, one of his earliest papers about a lottery problem, namely the distribution of the number of draws needed for all 90 numbers to appear at least once. His (correct) solution is an alternating sum whose derivation proved a numerical challenge because its terms keep growing.

Laplace came up with a good and manageable approximation 30 years later (detailed in Appendix Two of the book). He also contributed to the demise of the Loterie by arguing against this “voluntary” tax on moral grounds, an opinion he shared with Talleyrand, a fellow traveler in perpetually and successfully adapting to changing political regimes.

It is a bit of a surprise to read that this rather profitable venture ended in 1836, more under bankers’ than moralists’ pressure. A new national lottery—based on printed tickets rather than bets on results—would be created a century later, in 1933, and survive the second World War, while the French Loto appearing in 1974 is a direct successor to Casanova’s lottery.

The book covers many fascinating aspects, from the daily run of the Loterie to the various measures taken (successfully) against fraudsters; to its survival during the Révolution and its extension through the (Napoleonic) Empire; to tests for fairness (thanks to data available from almanacs); to the behavior of bettors and the sale of “helping” books; to (Daniel) Bernoulli, Buffon, Condorcet, and Laplace modeling rewards and supporting decreasing marginal utility.

Note that there are hardly any mathematical formulas in this book, except for an appendix on the probabilities of wins and the returns, as well as Laplace’s (and Legendre’s) derivations. This makes the book eminently suited for a large audience, the more thanks to Stigler’s perfect style.

This (paperback) book is also attractively designed by the University of Chicago Press, with a pleasant font (Adobe Caslon Pro) and a nice cover involving Laplace undercover, taken from a painting owned by the author. The many reproductions of epoch documents are well-rendered and easily readable. And, needless to say, given Stigler’s scholarship, the annotated reference list is impressive. The book is thus standing witness to the remarkable skills of the author, who searched for related material for more than 30 years, from Parisian specialized booksellers to French, English, and American archives. He manages to bring a wealth of connections and characters into the story, such as Voltaire’s scheme to take advantage of an earlier French state lottery aimed at reimbursing state debtors. (Voltaire actually made a fortune of several millions francs out of its poor design.)

For my personal edification, the book also gives life to several Paris Métro stations, such as Pereire and Duverney. But the book’s contents will prove fascinating for an audience far larger than Parisian locals and Francophiles. Enjoy!

Bayes Factors for Forensic Decision Analyses with R

Silvia Bozza, Franco Taroni, and Alex Biedermann

Hardcover: 199 pages
Publisher: Springer (2022)
ISBN-13: 978-3031098383

This book reminds me of a private tutee to whom I was teaching mathematics in the 1980s and who once showed up with a reproduction of the Loto wheel he had bought to increase his chances of winning. This proved to be his last lesson with me!

The book provides a sort of blueprint for using Bayes factors in forensics for both investigative and evaluative purposes, including R code and free access to the whole material. I am, of course, unable to judge the relevance of the approach for forensic science (I was under the vague impression that Bayesian arguments were usually not well-received in the courtroom), but feel that, overall, the approach is rather one of repositioning the standard Bayesian tools in a forensic framework.

The (evaluative) purpose is to assign a value to the result of a comparison between an item of unknown source and an item from a known source.

Therefore, I found nothing shocking or striking in this standard presentation of Bayes factors, including the call to loss functions, if a bit overly expansive in its exposition. The style is also classical, with a choice of gray background for vignettes of R coding parts that we also picked for our R books. If anything, I would have expected more realistic discussions and illustrations of prior specification across the hypotheses (p. 34), while the authors mostly focus on conjugate priors and the (de Finetti) trick of the equivalent prior sample size.

Bayes factors are mostly assessed using a conservative version of Jeffreys’s “scale of evidence.” The computational section of the book introduces MCMC (briefly) and mentions importance sampling, harmonic mean (with a minimalist warning), and Chib’s formula (with no warning whatsoever).

The (investigative) purpose is to provide information in investigative proceedings. The scientist is meant to use the findings to generate hypothesis and suggestions for explanations of observations to give guidance to investigators or litigants.

Chapter 2 is about standard models: inferring about a proportion, with some Monte Carlo illustration, and the complication of background elements, and normal mean, with an improper prior making an appearance (p. 69), but no mention of the general prohibition against such generalized priors when using Bayes factors or even of the Lindley-Jeffreys paradox. Again, the main difference from Bayesian textbooks stands with the chosen examples.

Chapter 3 focuses on evidence evaluation (not in the computational sense) but, again, the coverage is about standard models: processing the binomial, multinomial, Poisson models, again through conjugates (with the side remark that Fig. 3.2 is rather unhelpful: When moving the prior probability of the null from zero to one, its posterior probability also moves from zero to one).

We are back to the normal mean case with the model variance being known, then unknown—an unintentionally funny remark (p. 96) about the dependence between mean and variance being seen as too restrictive and replaced with independence.

At last (for me!), the book points out (p. 99) that the Bayes factor is highly sensitive to the choice of the prior variance (Oh, Lindley-Jeffreys paradox, where art thou?), but with a return of the improper prior (on said variance, p. 102) with no debate about the ensuing validity of the Bayes factor.

Multivariate normal settings are also discussed, with Wishart priors on the precision matrix, and more details about Chib’s estimate of the evidence. This chapter also contains illustrations of the so-called score-based Bayes factor, which is simply (?) a Bayes factor using a distribution on a distance summary (between an hypothetical population and the data) and an approximation of the distributions of these summaries, provided enough data is available. I also spotted a potentially interesting foray into Bayes factor variability (Section 3.4.2), although not reaching all the way to a notion of Bayes factor posterior distributions.

Chapter 4 presents Bayes factors for investigation, where alternative(s) is(are) less specified, as testing for Basmati rice vs. non-Basmati rice. But the book does not consider any non-parametric alternative. All in all, it looks to me rather similar to Chapter 3—being back to binomial, multinomial models, with more discussions of prior specification, more normal, or non-normal model, where the prior distribution is puzzlingly estimated by a kernel density estimator, a portmanteau alternative (p. 157), more multivariate normals with Wishart priors, and an entry about classification and discrimination.

Bayesian Probability for Babies

Chris Ferrie and Dr. Sarah Kaiser

Hardcover: 24 pages
Publisher: Sourcebooks Explore (2019)
ISBN-13: 978-1492680796

My friend E.J. Wagenmakers sent me a copy of his Bayesian Thinking for Toddlers, with the selling line “a must-have for any toddler with even a passing interest in Ockham’s razor and the prequential principle.”

E.J. wrote the 43-page story and Viktor Beekman (of JASP sticker fame) drew the illustrations, including 43 dinosaurs. The book can be read for free but cannot be purchased because publishers were not interested at the time and self-publishing was not available at a high-enough quality level. Hence, in the end, only 200 copies were printed as JASP (E.J.’s software) material, with me being the happy owner of one of these.

The story follows two young girls competing for dinosaur expertise and being rewarded by cookies, in proportion to the probability of providing the correct answer to two dinosaur questions. Toddlers may get less enthusiastic than grown-ups about the message, but they will love the drawings (and the questions, if they are into dinosaurs).

This reminded me of the Bayesian Probability for Babies book by Chris Ferrie, which details the computation of the probability that a cookie contains candy when the first bite holds none. It is more genuinely intended for young kids, in shape and design, as can be checked on a YouTube video, with a hypothetical population of cookies (with and without candy) being the proxy for the prior distribution.

I hope no baby will be traumatized by being exposed too soon to the notions of prior and posterior. Only data can tell, 20 years from now, if the book induced a spike or a collapse in the proportion of Bayesian statisticians.

About the Author

Christian Robert is a professor of statistics at both the Université Paris-Dauphine PSL and University of Warwick, and an ERC-Synergy grant holder. He has authored eight books and more than 150 papers about applied probability, Bayesian statistics, and simulation methods. Currently deputy editor of Biometrika, Robert also served as co-editor of the Journal of the Royal Statistical Society Series B and as associate editor for most major statistical journals. He is a Fellow of the Institute of Mathematical Statistics, American Statistical Association, and International Society for Bayesian Analysis, and an IMS Medallion Lecturer.

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