## law of small numbers

Posted in Statistics with tags , , , , , , , on June 22, 2022 by xi'an

## secrets of the surface

Posted in Statistics with tags , , , , on October 30, 2020 by xi'an

## Martin Hairer gets Breakthrough Prize (and \$3M)

Posted in Books, University life with tags , , , , , , , , , on September 14, 2020 by xi'an

Just heard the news that Fields Medallist Martin Hairer (formerly U of Warwick) got the 2021 Breakthrough Prize in Mathematics for his unification theory of stochastic partial differential equations, which he likens to a form of Taylor expansion in the massive Inventiones paper describing this breakthrough. (Looking at the previous winners of the prize, who also made its selection committee, this represents a break from focussing primarily on algebraic geometry! If not from sticking to male recipients…)

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semi-linear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $\Phi^4_ 3$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.

## introduction à la Statistique, by Cédric Villani

Posted in Books, Kids, Statistics, University life with tags , , , , , , on January 26, 2014 by xi'an

On Tuesday, there was a series of talks (in French) celebrating Statistics, with an introduction by Cédric Villani. (The talks are reproduced on the French Statistical Society (SFDS) webpage.) Rather unpredictably (!), Villani starts from an early 20th Century physics experiment leading to the estimation of the Avogadro constant from a series of integers. (Repeating an earlier confusion of his, he substitutes the probability of observing a rare event under the null with the probability of the alternative on the Higgs boson to be true!) A special mention to/of Francis Galton’s “supreme law of unreason”. And of surveys, pointing out the wide variability of a result for standard survey populations. But missing the averaging and more statistical effect of accumulating surveys, a principle at the core of Nate Silver‘s predictions. A few words again about the Séralini et al. experiments on Monsanto genetically modified maize NK603, attacked for their lack of statistical foundations. And then, hear hear!, much more than a mere mention of phylogenetic inference, with explanations about inverse inference, Markov Chain Monte Carlo algorithms on trees, convergence of Metropolis algorithms by Persi Diaconis, and Bayesian computations! Of course, this could be seen more as numerical probability than as truly statistics, but it is still pleasant to hear.

The last part of the talk more predictably links Villani’s own field of optimal transportation (which I would translate as a copula problem…) and statistics, mostly understood as empirical distributions. I find it somewhat funny that Sanov’s theorem is deemed therein to be a (or even the) Statistics theorem! I wonder how many statisticians could state this theorem… The same remark applies for the Donsker-Varadhan theory of large deviations. Still, the very final inequality linking the three types of information concepts is just… beautiful! You may spot in the last minute a micro confusion in repeating twice the definition for Fisher’s information rather than deducing that the information associated with a location family is constant. (And a no-so-necessary mention of the Cramer-Rao bound on unbiased estimators. Which could have been quoted as the Fréchet-Darmois-Cramer-Rao bound in such historical grounds ) A pleasant moment, all in all! (There are five other talks on that page, including one by Emmanuel Candés.)

## Nobel prize in statistics???

Posted in Kids, Statistics, University life with tags , , , , on January 4, 2014 by xi'an

Xiao-Li Meng asked this question in his latest XL column, to which Andrew replied faster than I. And in the same mood as mine. I had taken part to a recent discussion on this topic within the IMS Council, namely whether or not the IMS should associate with other organisations like ASA towards funding and supporting this potential prize. My initial reaction was one of surprise that we could consider mimicking/hijacking the Nobel for our field. First, I dislike the whole spirit of most prizes, from the personalisation to the media frenzy and distortion, to the notion that we could rank discoveries and research careers within a whole field. And separate what is clearly due to a single individual from what is due to a team of researchers.

Being clueless about those fields, I will not get into a discussion of who should have gotten a Nobel Prize in medicine, physics, or chemistry. And who should not have. But there are certainly many worthy competitors to the actual winners. And this is not the point: I do not see how any of this fights the downfall of scientific students in most of the Western World. That is, how a teenager can get more enticed to undertake maths or physics studies because she saw a couple old guys wearing weird clothes getting a medal and a check in Sweden. I have no actual data, but could Xiao-Li give me a quantitative assessment of the fact that Nobel Prizes “attract future talent”? Chemistry departments keep closing for lack of a sufficient number of students, (pure) maths and physics departments threatened with the same fate… Even the Fields Medal, which has at least the appeal of being delivered to younger researchers, does not seem to fit Xiao-Li’s argument. (To take a specific example: The recent Fields medallist Cédric Villani is a great communicator and took advantage of his medal to promote maths throughout France, in conferences, the medias, and by launching all kinds of initiative. I still remain sceptical about the overall impact on recruiting young blood in maths programs [again with no data to back up my feeling).) I will even less mention Nobel prizes for literature and peace, as there clearly is a political agenda in the nomination. (And selecting Sartre for the Nobel prize for literature definitely discredited it. At least for me.)

“…the media and public have given much more attention to the Fields Medal than to the COPSS Award, even though the former has hardly been about direct or even indirect impact on everyday life.” XL

Well, I do not see this other point of Xiao-Li’s. Nobel prizes are not prestigious for their impact on society, as most people do not understand at all what the rewarded research (career) is about. The most extreme example is the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel: On the one hand, Xiao-Li is right in pointing out that this is a very successful post-Alfred creation of a “Nobel Prize”. On the other hand, the fact that some years see two competing theories simultaneously win leads me to consider that this prize gives priority to theoretical construct above any impact on the World’s economy. Obviously, this statement is a bit of shooting our field in the foot since the only statisticians who got a Nobel Prize are econometricians and game-theorists! Nonetheless, it also shows that the happy few statisticians who entered the Nobel Olympus did not bring a bonus to the field… I am thus  remaining my usual pessimistic self on the impact of a whatever-company Prize in Statistical Sciences in Memory of Alfred Nobel.

Another remark is the opposition between the COPSS Award, which remains completely ignored by the media (despite a wealth of great nominees with various domains of achievements) and the Fields Medal (which is not ignored). This has been a curse of Statistics that has been discussed at large, namely the difficulty to separate what is math and what is outside math within the field. The Fields Medal is clearly very unlikely to nominate a statistician, even a highly theoretical statistician, as there will always be “sexier” maths results, i.e. corpora of work that will be seen as higher maths than, say, the invention of the Lasso or the creation of generalized linear models. So there is no hope to reach for an alternative Fields Medal with the same shine. Just like the Nobel Prize.

Other issues I could have mentioned, but for the length of the current rant, are the creation of rewards for solving a specific problem (as some found in Machine Learning), for involving multidisciplinary and multicountry research teams, and for reaching new orders of magnitude in processing large data problems.