Terry Tao on Bayes… and Trump

“From the perspective of Bayesian probability, the grade given to a student can then be viewed as a measurement (in logarithmic scale) of how much the posterior probability that the student’s model was correct has improved over the prior probability.” T. Tao, what’s new, June 1

Jean-Michel Marin pointed out to me the recent post of Terry Tao on setting a subjective prior for allocating partial credits to multiple answer questions. (Although I would argue that the main purpose of multiple answer questions is to expedite grading!) The post considers only true-false questionnaires and the case when the student produces a probabilistic assessment of her confidence in the answer. In the format of a probability p for each question. The goal is then to devise a grading principle, f, such that f(p) goes to the right answer and f(1-p) to the wrong answer. This sounds very much like scoring weather forecasters and hence designing proper scoring rules. Which reminds me of the first time I heard a talk about this: it was in Purdue, circa 1988, and Morrie DeGroot gave a talk on scoring forecasters, based on a joint paper he had written with Susie Bayarri. The scoring rule is proper if the expected reward leads to pick p=q when p is the answer given by the student and q her true belief. Terry Tao reaches the well-known conclusion that the grading function f should be f(p)=log²(2p) where log² denotes the base 2 logarithm. One property I was unaware of is that the total expected score writes as N+log²(L) where L is the likelihood associated with the student’s subjective model. (This is the only true Bayesian aspect of the problem.)

An interesting and more Bayesian last question from Terry Tao is about what to do when the probabilities themselves are uncertain. More Bayesian because this is where I would introduce a prior model on this uncertainty, in a hierarchical fashion, in order to estimate the true probabilities. (A non-informative prior makes its way into the comments.) Of course, all this leads to a lot of work given the first incentive of asking multiple choice questions…

One may wonder at the link with scary Donald and there is none! But the next post by Terry Tao is entitled “It ought to be common knowledge that Donald Trump is not fit for the presidency of the United States of America”. And unsurprisingly, as an opinion post, it attracted a large number of non-mathematical comments.

Le Monde sans puzzle #933

While Le Monde mathematical puzzle is purely geometric this week

If twelve points in a plane are such that, for any 5-uplet of those, at least 4 are on the same circle, and if M is the largest number of those points on the same circle, what is the minimal value of M?

and not straightforward to solve with an R code, there are several entries of interest in the Sciences and Medicine leaflet. One about capture-mark-recapture: making fun of a PLOS One paper on a capture-recapture study about the movements of bed bugs in New Jersey apartments. Another one on the resolution by Terry Tao of Erdös’ discrepancy conjecture. Which states that. for any (deterministic) sequence f:N→{−1,+1} taking values in {−1,+1}, the discrepancy of f is infinite, when the discrepancy is defined as

The entry in Le Monde tells the story of the derivation of the result and in particular the role of the Polymath5 project launched by Tao. It is interesting it is such a hard problem when considering the equivalent for a random sequence, which is more or less the gambler’s ruin result of Huygens. And a third entry on the explosion of the predatory journals, which publish essentially every submission in open access provided the authors accept to pay “charges”. And borrow titles and formats from existing reviews to a point where they can fool authors…

Mathematics on public radio

Following the Field medals awarded to Cédric Villani (the head of IHP and a former Dauphine PhD, as our president quickly communicated about) and Ngô Bao Châu (formely at Orsay, now moving to Chicago a fact omitted on the local news!), the French public radio, France Inter, dedicated one of its evening phone debates (Le Telephone Sonne) to research in Mathematics. The panelists were Cédric Villani, Jean-Pierre Bourguignon (head of IHES), and Guy Métivier (head of Mathematics at CNRS). This actually happens rather often (i.e. more often than once every four years when a French mathematician gets a Field medal) and I came across several of those by happenstance. The exercise (explaining to the layman what is the meaning of doing research in Mathematics and what are the advances made by the medalists in the current case) is interesting if delicate as the temptations to get lyrical or technical abound! I found the panelists did a very good job last Wednesday night. (Note that Terry Tao gave a summary of the four panelists’ achievements on his blog.) A very sweet (and my preferred) question was whether or not mathematical objects/concepts have an existence of their own or are created in mathematicians’ minds. In linking pure and applied maths, ie stating that there was no frontier between the two and that pure mathematicians are often involved in fruitful applications, Cédric Villani even mentioned Statistics, which is nice even though I doubt there is the remotest chance a statistician could get the Field Medal. (Incidentally, the COPSS Award, which is the Field Medal of Statistics, was attributed to David Dunson, from Duke, last August. Well-deserved, David!)