Next May 1-3, I will attend the 4th Bayesian, Fiducial and Frequentist Conference at Harvard University (hopefully not under snow at that time of year), which is a meeting between philosophers and statisticians about foundational thinking in statistics and inference under uncertainty. This should be fun! (Registration is now open.)
Archive for the Books Category
This is the
fifth sixth volume of Ben Aaronovitch’s Rivers of London series. Which features PC Peter Grant from the London’s Metropolitan Police specialising in paranormal crime. Joining a line of magicians that was started by Isaac Newton. And with the help of water deities. Although this English magic sleuthing series does not compare with the superlative Jonathan Strange & Mr. Norrell single book, The Hanging Tree remains highly enjoyable, maybe more for its style and vocabulary than for the detective story itself, which does not sound completely coherent (unless I read it too quickly during the wee hours in Banff last week). And does not bring much about this part of London. Still a pleasure to read as the long term pattern of Aaronovitch’s universe slowly unravels and some characters get more substance and depth.
While visiting Warwick last week, Jean-Michel Marin pointed out and forwarded me this remarkable paper of Jerzy Neyman, published in 1937, and presented to the Royal Society by Harold Jeffreys.
“Leaving apart on one side the practical difficulty of achieving randomness and the meaning of this word when applied to actual experiments…”
“It may be useful to point out that although we are frequently witnessing controversies in which authors try to defend one or another system of the theory of probability as the only legitimate, I am of the opinion that several such theories may be and actually are legitimate, in spite of their occasionally contradicting one another. Each of these theories is based on some system of postulates, and so long as the postulates forming one particular system do not contradict each other and are sufficient to construct a theory, this is as legitimate as any other. “
This paper is fairly long in part because Neyman starts by setting Kolmogorov’s axioms of probability. This is of historical interest but also needed for Neyman to oppose his notion of probability to Jeffreys’ (which is the same from a formal perspective, I believe!). He actually spends a fair chunk on explaining why constants cannot have anything but trivial probability measures. Getting ready to state that an a priori distribution has no meaning (p.343) and that in the rare cases it does it is mostly unknown. While reading the paper, I thought that the distinction was more in terms of frequentist or conditional properties of the estimators, Neyman’s arguments paving the way to his definition of a confidence interval. Assuming repeatability of the experiment under the same conditions and therefore same parameter value (p.344).
“The advantage of the unbiassed [sic] estimates and the justification of their use lies in the fact that in cases frequently met the probability of their differing very much from the estimated parameters is small.”
“…the maximum likelihood estimates appear to be what could be called the best “almost unbiassed [sic]” estimates.”
It is also quite interesting to read that the principle for insisting on unbiasedness is one of producing small errors, because this is not that often the case, as shown by the complete class theorems of Wald (ten years later). And that maximum likelihood is somewhat relegated to a secondary rank, almost unbiased being understood as consistent. A most amusing part of the paper is when Neyman inverts the credible set into a confidence set, that is, turning what is random in a constant and vice-versa. With a justification that the credible interval has zero or one coverage, while the confidence interval has a long-run validity of returning the correct rate of success. What is equally amusing is that the boundaries of a credible interval turn into functions of the sample, hence could be evaluated on a frequentist basis, as done later by Dennis Lindley and others like Welch and Peers, but that Neyman fails to see this and turn the bounds into hard values. For a given sample.
“This, however, is not always the case, and in general there are two or more systems of confidence intervals possible corresponding to the same confidence coefficient α, such that for certain sample points, E’, the intervals in one system are shorter than those in the other, while for some other sample points, E”, the reverse is true.”
The resulting construction of a confidence interval is then awfully convoluted when compared with the derivation of an HPD region, going through regions of acceptance that are the dual of a confidence interval (in the sampling space), while apparently [from my hasty read] missing a rule to order them. And rejecting the notion of a confidence interval being possibly empty, which, while being of practical interest, clashes with its frequentist backup.
Just heard the great news that the Abel Prize for 2017 goes to Yves Meyer! Yves Meyer is an emeritus professor at École Normale de Cachan and has produced fundamental contributions to number theory, operator theory and harmonic analysis. He is one of the originators of the theory of wavelets and multiresolution analysis. Among other recognitions and prizes, he was an invited speaker at the International Congress of Mathematicians in 1970 (Nice), in 1983 (Warsaw), and in 1990 (Kyoto), and was awarded the Gauß Prize in 2010. Congratulations and total respect to Yves Meyer!!!
This 2016 book by Howard Wainer has been sitting (!) on my desk for quite a while and it took a long visit to Warwick to find a free spot to quickly read it and write my impressions. The subtitle is, as shown on the picture, “Distinguishing fact from fiction by learning to think like a data scientist”. With all due respect to the book, which illustrates quite pleasantly the dangers of (pseudo-)data mis- or over- (or eve under-)interpretation, and to the author, who has repeatedly emphasised those points in his books and
tribunes opinion columns, including those in CHANCE, I do not think the book teaches how to think like a data scientist. In that an arbitrary neophyte reader would not manage to handle a realistic data centric situation without deeper training. But this collection of essays, some of which were tribunes, makes for a nice reading nonetheless.
I presume that in this post-truth and alternative facts [dark] era, the notion of truthiness is familiar to most readers! It is often based on a misunderstanding or a misappropriation of data leading to dubious and unfounded conclusions. The book runs through dozens of examples (some of them quite short and mostly appealing to common sense) to show how this happens and to some extent how this can be countered. If not avoided as people will always try to bend, willingly or not, the data to their conclusion.
There are several parts and several themes in Truth or Truthiness, with different degrees of depth and novelty. The more involved part is in my opinion the one about causality, with illustrations in educational testing, psychology, and medical trials. (The illustration about fracking and the resulting impact on Oklahoma earthquakes should not be in the book, except that there exist officials publicly denying the facts. The same remark applies to the testing cheat controversy, which would be laughable had not someone ended up the victim!) The section on graphical representation and data communication is less exciting, presumably because it comes after Tufte’s books and message. I also feel the 1854 cholera map of John Snow is somewhat over-exploited, since he only drew the map after the epidemic declined. The final chapter
Don’t Try this at Home is quite anecdotal and at the same time this may the whole point, namely that in mundane questions thinking like a data scientist is feasible and leads to sometimes surprising conclusions!
“In the past a theory could get by on its beauty; in the modern world, a successful theory has to work for a living.” (p.40)
The book reads quite nicely, as a whole and a collection of pieces, from which class and talk illustrations can be borrowed. I like the “learned” tone of it, with plenty of citations and witticisms, some in Latin, Yiddish and even French. (Even though the later is somewhat inaccurate! Si ça avait pu se produire, ça avait dû se produire [p.152] would have sounded more vernacular in my Gallic opinion!) I thus enjoyed unreservedly Truth or Truthiness, for its rich style and critical message, all the more needed in the current times, and far from comparing it with a bag of potato chips as Andrew Gelman did, I would like to stress its classical tone, in the sense of being immersed in a broad and deep culture that seems to be receding fast.
After discussing Jonathan Strange & Mr Norrell with David Frazier in Banff, where I spotted him reading this fabulous book, I went for a look at the series BBC One made out of this great novel. And got so hooked to it that I binge-watched the whole series of 7 episodes over three days..! I am utterly impressed at the BBC investing so much into this show, rendering most of the spirit of the book and not only the magical theatrics. The complex [and nasty] personality of Mr Norrell and his petit-bourgeois quest of respectability is beautifully exposed, leading him to lie and steal and come close to murder [directly or by proxy], in a pre-Victorian and anti-Romantic urge to get away from magical things from the past, “more than 300 years ago”. While Jonathan Strange’s own Romantic inclinations are obvious, including the compulsory travel to Venezia [even though the BBC could only afford Croatia, it seems!] The series actually made clear some points I had missed in the novel, presumably by rushing through it, like the substitution of Strange’s wife by the moss-oak doppelganger created by the fairy king. The enslavement of Stephen, servant of Lord Pole and once and future king by the same fairy is also superbly rendered.
While not everything in the series is perfect, with in particular the large scale outdoor scenes being too close to a video-game rendering (as in the battle of Waterloo that boils down to a backyard brawl!), the overall quality of the show [the Frenchmen there parlent vraiment français, with no accent!] and adhesion to the spirit of Susanna Clarke’s novel make it an example of the tradition of excellence of the BBC. (I just wonder at the perspective of a newcomer who would watch the series with no prior exposure to the book!)
In a simple question on X validated a few days ago [about simulating from x²φ(x)] popped up the remark that the person asking the question wanted a direct simulation method for higher efficiency. Compared with an accept-reject solution. Which shows a misunderstanding of what “efficiency” means on Monte Carlo situations. If it means anything, I would think it is reflected in the average time taken to return one simulation and possibly in the worst case. But there is no reason to call an inverse cdf method more efficient than an accept reject or a transform approach since it all depends on the time it takes to make the inversion compared with the other solutions… Since inverting the closed-form cdf in this example is much more expensive than generating a Gamma(½,½), and taking plus or minus its root, this is certainly the case here. Maybe a ziggurat method could be devised, especially since x²φ(x)<φ(x) when |x|≤1, but I am not sure it is worth the effort!