Archive for integration

the amazing adventures of Kavalier & Clay [book review]

Posted in Books, Kids, Travel with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on August 2, 2024 by xi'an

After reading the (pretty good) Yiddish Policemen’s Union by Michael Chabon, I found out this earlier book had won a Pullitzer prize for fiction and decided to give it a go. Which lasted for weeks as I found the book very slow paced, to the point of always falling asleep within a few pages. The pace sort of accelerated in the second half, sort of coïnciding with Pearl Harbour and the formal entry of the United States of America into the war. At the beginning it felt too much like a transposition of my former read and despite the multiple covariates with a positive coefficient (comics, Prague, the golem, Rosa Luxemburg, New York City, the silliness of the superhero concept, esp. when repeated week after week, the ambivalence of the very notion as well, when considering the closeness between Supermen and Hitler’s Übermensch), I had a hard time engaging with the story. Maybe due to the lengthy sentences and descriptions which, as I read later in a review, were intended to supplement the absence of visual representations that (obviously) come with a graphical novel. Maybe due to the millefeuille of literary styles and fields and of parallel stories with a myriad of characters and places.

“Besotted with language and brimming with pop culture, political relevance and bravura storytelling…” The New York Times

I clearly am a minority in enjoying the book so little. The Guardian put it within its 100 best books of the 21st Century. If in the second half.  And so did the New York Times. In its 16th position. There is even a (inactive) website on the book! Actually, the massive injection within the actual history of America and Europa at these most tragic times makes the novel an epic, helped with a witty and mostly understated humour. There are indeed lots of actual facts and characters, the book deeply digs into the history of the time. While indirect snapshots of the unrolling Holocaust are ghastly,  Both the sinking of the orphan boat (presumably inspired by the all too real attack on the City of Benares) and the Antarctica part during WW II are sheer fiction (despite a long-going myth of Nazi bases there). But a compelling part of the novel, paradoxically.

Eeeech… [X validated]

Posted in Books, Statistics, University life with tags , , , , on November 3, 2023 by xi'an

why is the likelihood not a pdf?

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , on January 4, 2021 by xi'an

The return of an old debate on X validated. Can the likelihood be a pdf?! Even though there exist cases where a [version of the] likelihood function shows such a symmetry between the sufficient statistic and the parameter, as e.g. in the Normal mean model, that they are somewhat exchangeable w.r.t. the same measure, the question is somewhat meaningless for a number of reasons that we can all link to Ronald Fisher:

  1. when defining the likelihood function, Fisher (in his 1912 undergraduate memoir!) warns against integrating it w.r.t. the parameter: “the integration with respect to m is illegitimate and has no definite meaning with respect to inverse probability”. The likelihood is “is a relative probability only, suitable to compare point with point, but incapable of being interpreted as a probability distribution over a region, or of giving any estimate of absolute probability.” And again in 1922: “[the likelihood] is not a differential element, and is incapable of being integrated: it is assigned to a particular point of the range of variation, not to a particular element of it”.
  2. He introduced the term “likelihood” especially to avoid the confusion: “I perceive that the word probability is wrongly used in such a connection: probability is a ratio of frequencies, and about the frequencies of such values we can know nothing whatever (…) I suggest that we may speak without confusion of the likelihood of one value of p being thrice the likelihood of another (…) likelihood is not here used loosely as a synonym of probability, but simply to express the relative frequencies with which such values of the hypothetical quantity p would in fact yield the observed sample”.
  3. Another point he makes repeatedly (both in 1912 and 1922) is the lack of invariance of the probability measure obtained by attaching a dθ to the likelihood function L(θ) and normalising it into a density: while the likelihood “is entirely unchanged by any [one-to-one] transformation”, this definition of a probability distribution is not. Fisher actually distanced himself from a Bayesian “uniform prior” throughout the 1920’s.

which sums up as the urge to never neglect the dominating measure!

computational methods for numerical analysis with R [book review]

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , , , , , , , , on October 31, 2017 by xi'an

compulysis+R_coverThis is a book by James P. Howard, II, I received from CRC Press for review in CHANCE. (As usual, the customary warning applies: most of this blog post will appear later in my book review column in CHANCE.) It consists in a traditional introduction to numerical analysis with backup from R codes and packages. The early chapters are setting the scenery, from basics on R to notions of numerical errors, before moving to linear algebra, interpolation, optimisation, integration, differentiation, and ODEs. The book comes with a package cmna that reproduces algorithms and testing. While I do not find much originality in the book, given its adherence to simple resolutions of the above topics, I could nonetheless use it for an elementary course in our first year classes. With maybe the exception of the linear algebra chapter that I did not find very helpful.

“…you can have a solution fast, cheap, or correct, provided you only pick two.” (p.27)

The (minor) issue I have with the book and that a potential mathematically keen student could face as well is that there is little in the way of justifying a particular approach to a given numerical problem (as opposed to others) and in characterising the limitations and failures of the presented methods (although this happens from time to time as e.g. for gradient descent, p.191). [Seeping in my Gallic “mal-être”, I am prone to over-criticise methods during classing, to the (increased) despair of my students!, but I also feel that avoiding over-rosy presentations is a good way to avoid later disappointments or even disasters.] In the case of this book, finding [more] ways of detecting would-be disasters would have been nice.

An uninteresting and highly idiosyncratic side comment is that the author preferred the French style for long division to the American one, reminding me of my first exposure to the latter, a few months ago! Another comment from a statistician is that mentioning time series inter- or extra-polation without a statistical model sounds close to anathema! And makes extrapolation a weapon without a cause.

“…we know, a priori, exactly how long the [simulated annealing] process will take since it is a function of the temperature and the cooling rate.” (p.199)

Unsurprisingly, the section on Monte Carlo integration is disappointing for a statistician/probabilistic numericist like me,  as it fails to give a complete enough picture of the methodology. All simulations seem to proceed there from a large enough hypercube. And recommending the “fantastic” (p.171) R function integrate as a default is scary, given the ability of the selected integration bounds to misled its users. Similarly, I feel that the simulated annealing section is not providing enough of a cautionary tale about the highly sensitive impact of cooling rates and absolute temperatures. It is only through the raw output of the algorithm applied to the travelling salesman problem that the novice reader can perceive the impact of some of these factors. (The acceptance bound on the jump (6.9) is incidentally wrongly called a probability on p.199, since it can take values larger than one.)

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE.]