Terry Tao on Bayes… and Trump

Posted in Books, Kids, Statistics, University life with tags , , , , , , , on June 13, 2016 by xi'an

“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

Posted in Books, Kids, Statistics, University life with tags , , , , , on October 17, 2015 by xi'an

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

$\sup_{n,d} \left|\sum_{j=1}^n f(jd)\right|$

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…