death notice from Bourbaki

a unified treatment of predictive model comparison

“Applying various approximation strategies to the relative predictive performance derived from predictive distributions in frequentist and Bayesian inference yields many of the model comparison techniques ubiquitous in practice, from predictive log loss cross validation to the Bayesian evidence and Bayesian information criteria.”

Michael Betancourt (Warwick) just arXived a paper formalising predictive model comparison in an almost Bourbakian sense! Meaning that he adopts therein a very general representation of the issue, with minimal assumptions on the data generating process (excluding a specific metric and obviously the choice of a testing statistic). He opts for an M-open perspective, meaning that this generating process stands outside the hypothetical statistical model or, in Lindley’s terms, a small world. Within this paradigm, the only way to assess the fit of a model seems to be through the predictive performances of that model. Using for instance an f-divergence like the Kullback-Leibler divergence, based on the true generated process as the reference. I think this however puts a restriction on the choice of small worlds as the probability measure on that small world has to be absolutely continuous wrt the true data generating process for the distance to be finite. While there are arguments in favour of absolutely continuous small worlds, this assumes a knowledge about the true process that we simply cannot gather. Ignoring this difficulty, a relative Kullback-Leibler divergence can be defined in terms of an almost arbitrary reference measure. But as it still relies on the true measure, its evaluation proceeds via cross-validation “tricks” like jackknife and bootstrap. However, on the Bayesian side, using the prior predictive links the Kullback-Leibler divergence with the marginal likelihood. And Michael argues further that the posterior predictive can be seen as the unifying tool behind information criteria like DIC and WAIC (widely applicable information criterion). Which does not convince me towards the utility of those criteria as model selection tools, as there is too much freedom in the way approximations are used and a potential for using the data several times.

the theory that would not die…

A few days ago, I had lunch with Sharon McGrayne in a Parisian café and we had a wonderful chat about the people she had met during the preparation of her book, the theory that would not die. Among others, she mentioned the considerable support provided by Dennis Lindley, Persi Diaconis, and Bernard Bru. She also told me about a few unsavoury characters who simply refused to talk to her about the struggles and rise of Bayesian statistics. Then, once I had biked home, her book had at last arrived in my mailbox! How timely! (Actually, getting the book before would have been better, as I would have been able to ask more specific questions. But it seems the publisher, Yale University Press, had not forecasted the phenomenal success of the book and thus failed to scale the reprints accordingly!)

 Here is thus my enthusiastic (and obviously biased) reaction to the theory that would not die. It tells the story and the stories of Bayesian statistics and of Bayesians in a most genial and entertaining manner. There may be some who will object to such a personification of science, which should be (much) more than the sum of the characters who contributed to it. However, I will defend the perspective that (Bayesian) statistical science is as much philosophy as it is mathematics and computer-science, thus that the components that led to its current state were contributed by individuals, for whom the path to those components mattered. While the book inevitably starts with the (patchy) story of Thomas Bayes’s life, incl. his passage in Edinburgh, and a nice non-mathematical description of his ball experiment, the next chapter is about “the man who did everything”, …, yes indeed, Pierre-Simon (de) Laplace himself! (An additional nice touch is the use of lower case everywhere, instead of an inflation of upper case letters!) How Laplace attacked the issue of astronomical errors is brilliantly depicted, rooting the man within statistics and explaining  why he would soon move to the “probability of causes”. And rediscover plus generalise Bayes’ theorem. That his (rather unpleasant!) thirst for honours and official positions would cause later disrepute on his scientific worth is difficult to fathom, esp. when coming from knowledgeable statisticians like Florence Nightingale David. The next chapter is about the dark ages of [not yet] Bayesian statistics and I particularly liked the links with the French army, discovering there that the great Henri Poincaré testified at Dreyfus’ trial using a Bayesian argument, that Bertillon had completely missed the probabilistic point, and that the military judges were then all aware of Bayes’ theorem, thanks to Bertrand’s probability book being used at École Polytechnique! (The last point actually was less of a surprise, given that I had collected some documents about the involvement of late 19th/early 20th century artillery officers in the development of Bayesian techniques, Edmond Lhostes and Maurice Dumas, in connection with Lyle Broemeling’s Biometrika study.) The description of the fights between Fisher and Bayesians and non-Bayesians alike is as always both entertaining and sad. Sad also is the fact that Jeffreys’ masterpiece got so little recognition at the time. (While I knew about Fisher’s unreasonable stand on smoking, going as far as defending the assumption that “lung cancer might cause smoking”(!), the Bayesian analysis of Jerome Cornfield was unknown to me. And quite fascinating.) The figure of Fisher actually permeates the whole book, as a negative bullying figure preventing further developments of early Bayesian statistics, but also as an ambivalent anti-Bayesian who eventually tried to create his own brand of Bayesian statistics in the format of fiducial statistics…

“…and then there was the ghastly de Gaulle.” D. Lindley

The following part of the theory that would not die is about Bayes’ contributions to the war (WWII), at least from the Allied side. Again, I knew most of the facts about Alan Turing and Bletchley Park’s Enigma, however the story is well-told and, as in previous occasions, I cannot but be moved by the waste of such a superb intellect, thanks to the stupidity of governments. The role of Albert Madansky in the assessment of the [lack of] safety of nuclear weapons is also well-described, stressing the inevitability of a Bayesian assessment of a one-time event that had [thankfully] not yet happened. The above quote from Dennis Lindley is the conclusion of his argument on why Bayesian statistics were not called Laplacean; I would think instead that the French post-war attraction for abstract statistics in the wake of Bourbaki did more against this recognition than de Gaulle’s isolationism and ghastliness. The involvement of John Tukey into military research was also a novelty for me, but not so much as his use of Bayesian [small area] methods for NBC election night previsions. (They could not hire José nor Andrew at the time.) The conclusion of Chapter 14 on why Tukey felt the need to distance himself from Bayesianism is quite compelling. Maybe paradoxically, I ended up appreciating Chapter 15 even more for the part about the search for a missing H-bomb near Palomares, Spain, as it exposes the plusses a Bayesian analysis would have brought.

“There are many classes of problems where Bayesian analyses are reasonable, mainly classes with which I have little acquaintance.” J. Tukey

When approaching near recent times and to contemporaries,  Sharon McGrayne gives a very detailed coverage of the coming-of-age of Bayesians like Jimmy Savage and Dennis Lindley, as well as the impact of Stein’s paradox (a personal epiphany!), along with the important impact of Howard Raiffa and Robert Schlaifer, both on business schools and on modelling prior beliefs [via conjugate priors]. I did not know anything about their scientific careers, but Applied Statistical Decision Theory is a beautiful book that prefigured both DeGroot‘s and Berger‘s. (As an aside, I was amused by Raiffa using Bayesian techniques for horse betting based on race bettors, as I had vaguely played with the idea during my spare if compulsory time in the French Navy!) Similarly, while I’d read detailed scientific accounts of Frederick Mosteller’s and David Wallace’s superb Federalist Papers study, they were only names to me. Chapter 12 mostly remedied this lack of mine’s.

“We are just starting” P. Diaconis

The final part, entitled Eureka!, is about the computer revolution we witnessed in the 1980’s, culminating with the (re)discovery of MCMC methods we covered in our own “history”. Because it contains stories that are closer and closer to today’s time, it inevitably crumbles into shorter and shorter accounts. However, the theory that would not die conveys the essential message that Bayes’ rule had become operational, with its own computer language and objects like graphical models and Bayesian networks that could tackle huge amounts of data and real-time constraints. And used by companies like Microsoft and Google. The final pages mention neurological experiments on how the brain operates in a Bayesian-like way (a direction much followed by neurosciences, as illustrated by Peggy Series’ talk at Bayes-250).

In conclusion, I highly enjoyed reading through the theory that would not die. And I am sure most of my Bayesian colleagues will as well. Being Bayesians, they will compare the contents with their subjective priors about Bayesian history, but will in the end update those profitably. (The most obvious missing part is in my opinion the absence of E.T Jaynes and the MaxEnt community, which would deserve a chapter on its own.) Maybe ISBA could consider supporting a paperback or electronic copy to distribute to all its members! As an insider, I have little idea on how the book would be perceived by the layman: it does not contain any formula apart from [the discrete] Bayes’ rule at some point, so everyone can read through.  The current success of the theory that would not die shows that it reaches much further than academic circles. It may be that the general public does not necessarily grasp the ultimate difference between frequentist and Bayesians, or between Fisherians and Neyman-Pearsonians. However the theory that would not die goes over all the elements that explain these differences. In particular, the parts about single events are quite illuminating on the specificities of the Bayesian approach. I will certainly [more than] recommend it to all of my graduate students (and buy the French version for my mother once it is translated, so that she finally understands why I once gave a talk “Don’t tell my mom I am Bayesian” at ENSAE…!) If there is any doubt from the above, I obviously recommend the book to all Og’s readers!