Archive for undergraduates

Bayes Rules! [book review]

Posted in Books, Kids, Mountains, pictures, R, Running, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on July 5, 2022 by xi'an

Bayes Rules! is a new introductory textbook on Applied Bayesian Model(l)ing, written by Alicia Johnson (Macalester College), Miles Ott (Johnson & Johnson), and Mine Dogucu (University of California Irvine). Textbook sent to me by CRC Press for review. It is available (free) online as a website and has a github site, as well as a bayesrule R package. (Which reminds me that both our own book R packages, bayess and mcsm, have gone obsolete on CRAN! And that I should find time to figure out the issue for an upgrading…)

As far as I can tell [from abroad and from only teaching students with a math background], Bayes Rules! seems to be catering to early (US) undergraduate students with very little exposure to mathematical statistics or probability, as it introduces basic probability notions like pmf, joint distribution, and Bayes’ theorem (as well as Greek letters!) and shies away from integration or algebra (a covariance matrix occurs on page 437 with a lot . For instance, the Normal-Normal conjugacy derivation is considered a “mouthful” (page 113). The exposition is somewhat stretched along the 500⁺ pages as a result, imho, which is presumably a feature shared with most textbooks at this level, and, accordingly, the exercises and quizzes are more about intuition and reproducing the contents of the chapter than technical. In fact, I did not spot there a mention of sufficiency, consistency, posterior concentration (almost made on page 113), improper priors, ergodicity, irreducibility, &tc., while other notions are not precisely defined, like ESS, weakly informative (page 234) or vague priors (page 77), prior information—which makes the negative answer to the quiz “All priors are informative”  (page 90) rather confusing—, R-hat, density plot, scaled likelihood, and more.

As an alternative to “technical derivations” Bayes Rules! centres on intuition and simulation (yay!) via its bayesrule R package. Itself relying on rstan. Learning from example (as R code is always provided), the book proceeds through conjugate priors, MCMC (Metropolis-Hasting) methods, regression models, and hierarchical regression models. Quite impressive given the limited prerequisites set by the authors. (I appreciated the representations of the prior-likelihood-posterior, especially in the sequential case.)

Regarding the “hot tip” (page 108) that the posterior mean always stands between the prior mean and the data mean, this should be made conditional on a conjugate setting and a mean parameterisation. Defining MCMC as a method that produces a sequence of realisations that are not from the target makes a point, except of course that there are settings where the realisations are from the target, for instance after a renewal event. Tuning MCMC should remain a partial mystery to readers after reading Chapter 7 as the Goldilocks principle is quite vague. Similarly, the derivation of the hyperparameters in a novel setting (not covered by the book) should prove a challenge, even though the readers are encouraged to “go forth and do some Bayes things” (page 509).

While Bayes factors are supported for some hypothesis testing (with no point null), model comparison follows more exploratory methods like X validation and expected log-predictive comparison.

The examples and exercises are diverse (if mostly US centric), modern (including cultural references that completely escape me), and often reflect on the authors’ societal concerns. In particular, their concern about a fair use of the inferred models is preminent, even though a quantitative assessment of the degree of fairness would require a much more advanced perspective than the book allows… (In that respect, Exercise 18.2 and the following ones are about book banning (in the US). Given the progressive tone of the book, and the recent ban of math textbooks in the US, I wonder if some conservative boards would consider banning it!) Concerning the Himalaya submitting running example (Chapters 18 & 19), where the probability to summit is conditional on the age of the climber and the use of additional oxygen, I am somewhat surprised that the altitude of the targeted peak is not included as a covariate. For instance, Ama Dablam (6848 m) is compared with Annapurna I (8091 m), which has the highest fatality-to-summit ratio (38%) of all. This should matter more than age: the Aosta guide Abele Blanc climbed Annapurna without oxygen at age 57! More to the point, the (practical) detailed examples do not bring unexpected conclusions, as for instance the fact that runners [thrice alas!] tend to slow down with age.

A geographical comment: Uluru (page 267) is not a city!, but an impressive sandstone monolith in the heart of Australia, a 5 hours drive away from Alice Springs. And historical mentions: Alan Turing (page 10) and the team at Bletchley Park indeed used Bayes factors (and sequential analysis) in cracking the Enigma, but this remained classified information for quite a while. Arianna Rosenbluth (page 10, but missing on page 165) was indeed a major contributor to Metropolis et al.  (1953, not cited), but would not qualify as a Bayesian statistician as the goal of their algorithm was a characterisation of the Boltzman (or Gibbs) distribution, not statistical inference. And David Blackwell’s (page 10) Basic Statistics is possibly the earliest instance of an introductory Bayesian and decision-theory textbook, but it never mentions Bayes or Bayesianism.

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Book Review section in CHANCE.]

hands-on probability 101

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , , on April 3, 2021 by xi'an


When solving a rather simple probability question on X validated, namely the joint uniformity of the pair

(X,Y)=(A-B+\mathbb I_{A<B},C-B+\mathbb I_{C<B})

when A,B,C are iid U(0,1), I chose a rather pedestrian way and derived the joint distribution of (A-B,C-B), which turns to be made of 8 components over the (-1,1)² domain. And to conclude at the uniformity of the above, I added a hand-made picture to explain why the coverage by (X,Y) of any (red) square within (0,1)² was uniform by virtue of the symmetry between the coverage by (A-B,C-B) of four copies of the (red) square, using color tabs that were sitting on my desk..! It did not seem to convince the originator of the question, who kept answering with more questions—or worse an ever-changing question, reproduced in real time on math.stackexchange!, revealing there that said originator was tutoring an undergrad student!—but this was a light moment in a dreary final day before a new lockdown.

a glaringly long explanation

Posted in Statistics with tags , , , , , , , , , , on December 19, 2018 by xi'an

It is funny that, when I am teaching the rudiments of Bayesian statistics to my undergraduate students in Paris-Dauphine, including ABC via Rasmus’ socks, specific questions about the book (The Bayesian Choice) start popping up on X validated! Last week was about the proof that ABC is exact when the tolerance is zero. And the summary statistic sufficient.

This week is about conjugate distributions for exponential families (not that there are many others!). Which led me to explain both the validation of the conjugacy and the derivation of the posterior expectation of the mean of the natural sufficient statistic in far more details than in the book itself. Hopefully in a profitable way.

done! [#1]

Posted in Kids, pictures, University life with tags , , , , , , on January 16, 2016 by xi'an

After spending a few hours grading my 127 exams for most nights of this week, I am finally done with it! One of the exam questions was the simulation of XY when (X,Y) is a bivariate normal vector with correlation ρ, following the trick described in a X validated question asked a few months ago, namely that

XY≡R{cos(πU)+ρ}

but no one managed to establish this representation. And, as usual, some students got confused between parameters θ and observations x when writing a posterior density, since the density of the prior was defined in the exam with the dummy x, thereby recovering the prior as the posterior. Nothing terrible and nothing exceptional with this cohort of undergraduates. And now I still have to go through my second pile of exams for the graduate course I taught on Bayesian computational tools…

Statistics second slides

Posted in Books, Kids, Statistics, University life with tags , , , , , on September 24, 2014 by xi'an

La Défense from Paris-Dauphine, Nov. 15, 2012This is the next chapter of my Statistics course, definitely more standard, with some notions on statistical models, limit theorems, and exponential families. In the first class, I recalled the convergence notions with no proof but counterexamples and spend some time on a slide not included here, borrowed from Chris Holmes’ talk last Friday on the linear relation between blood pressure and the log odds ratio of an heart condition. This was a great example, both to illustrate the power of increasing the number of observations and of using a logistic regression model. Students kept asking questions about it.

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