folded Normals

Posted in Books, Kids, pictures, R, Running, Statistics with tags , , , , , , , , , , , , on February 25, 2021 by xi'an

While having breakfast (after an early morn swim at the vintage La Butte aux Cailles pool, which let me in free!), I noticed a letter to the Editor in the Annals of Applied Statistics, which I was unaware existed. (The concept, not this specific letter!) The point of the letter was to indicate that finding the MLE for the mean and variance of a folded normal distribution was feasible without resorting to the EM algorithm. Since the folded normal distribution is a special case of mixture (with fixed weights), using EM is indeed quite natural, but the author, Iain MacDonald, remarked that an optimiser such as R nlm() could be called instead. The few lines of relevant R code were even included. While this is a correct if minor remark, I am a wee bit surprised at seeing it included in the journal, the more because the authors of the original paper using the EM approach were given the opportunity to respond, noticing EM is much faster than nlm in the cases they tested, and Iain MacDonald had a further rejoinder! The more because the Wikipedia page mentioned the use of optimisers much earlier (and pointed out at the R package Rfast as producing MLEs for the distribution).

MCMC for conditional Bernoullis

Posted in Books, Statistics, University life with tags , , , , , on February 22, 2021 by xi'an

Jeremy Heng, Pierre Jacob [currently in Paris!] and Nianqiao Ju are in a recent arXival considering the simulation of a conditional Bernoulli, namely generating a vector of N Bernoullis with different probabilities under the constraint that their sum is fixed as I. Rather than going for a perfect simulator, with cost O(NI), they opt for the simplest of MCMC samplers, where a 0 and a 1 entries are exchanged at random. In connection with a recent spate of MCMC works using coupling, they establish convergence in O(N log N) steps, even when the probabilities are arbitrarily close to zero and one. Including the case when they are Uniformly generated. From a mundane perspective, I wonder at the appeal of using the probabilities to select the exchange pair. I realise sorting the probabilities is already of order O(N log N) avoiding selecting highly probable 1’s and highly probable 0’s should speed up converge, unless the gain is negligible. And to link MCMC and exact simulation in this setting, what would the cost of perfect sampling by sampling from the past be? Presumably much higher since there is little chance a total ordering can be found on the starting states.

Posted in Books, Kids with tags , , , , , , on February 20, 2021 by xi'an

After seeing the books in a Denver bookstore (in the summer of 2019), I eventually came to try one, then the others. Even though the setting is somewhat simplistic, or may intended for young adults, with ideas also found in earlier novels, it makes for a pleasant read. The underlying concept is having several Londons set in different universes and connected by magic for the happy few able to travel between them. One of them is “our” Victorian London. Labelled as Grey London. Then there are White, Red, and Black Londons… With some pivotal pubs existing in all (?) of them. This reminded me very much of Neverwhere, one of the few Gaimans I deeply appreciated. Or of Pullman’s Oxfords. The first volume sets the scene, with two main characters, (Grey) Lila and (Red) Kell, whose paths will come to cross, some villains in the least privileged London, and some sudden existential threat on Red London. The latest being the least convincing part of the plot as lacking subtlety. The second volume mostly takes place in Red London and the first part sounds a wee bit like the female part of Red seas under red skys. That is, a smart thief at sea. And a smarter captain. With on top of it a magic competition where all main characters cross path. Again a poor part of the plot as the competition feels like Harry Potter’s Goblet of Fire, while a new danger is building up to bring the fodder for the third volume. Not completely uninteresting (I read most of it over Xmas day, by a log fire), but somewhat two-dimensional (with a surprising lack of moral reticence to kill people, most surprising for a YA series). The third volume, A conjuring of light,  is a bit more predictable, including the deaths of some major characters (one or two more would have helped). And the ending could have been less all-inclusive and rosy!,  but this was an enjoyable conclusion nonetheless.

the Ramanujan machine

Posted in Books, Kids, pictures, University life with tags , , , , , , , , , , , on February 18, 2021 by xi'an

Nature of 4 Feb. 2021 offers a rather long (Nature-like) paper on creating Ramanujan-like expressions using an automated process. Associated with a cover in the first pages. The purpose of the AI is to generate conjectures of Ramanujan-like formulas linking famous constants like π or e and algebraic formulas like the novel polynomial continued fraction of 8/π²:

$\frac{8}{{{\rm{\pi }}}^{2}}=1-\frac{2\times {1}^{4}-{1}^{3}}{7-\frac{2\times {2}^{4}-{2}^{3}}{19-\frac{2\times {3}^{4}-{3}^{3}}{37-\frac{2\times {4}^{4}-{4}^{3}}{\ldots }}}}$

which currently remains unproven. The authors of the “machine” provide Python code that one can run to try uncover new conjectures, possibly named after the discoverer! The article is spending a large proportion of its contents to justify the appeal of generating such conjectures, with several unsuspected formulas later proven for real, but I remain unconvinced of the deeper appeal of the machine (as well as unhappy about the association of Ramanujan and machine, since S. Ramanujan had a mystical and unexplained relation to numbers, defeating Hardy’s logic,  “a mathematician of the highest quality, a man of altogether exceptional originality and power”). The difficulty is in separating worthwhile from anecdotal (true) conjectures, not to mention wrng conjectures. This is certainly of much deeper interest than separating chihuahua faces from blueberry muffins, but does it really “help to create mathematical knowledge”?

monomial representations on Netflix

Posted in Books, Kids, pictures, Travel with tags , , , , , , , , , , , , on February 16, 2021 by xi'an

When watching the first episode of Queen’s Gambit, following the recommendations of my son, I glimpsed the cover of a math thesis defended at Cornell by the mother of the main character..! Prior to 1957, year of her death. Searching a wee bit further, I found that there exists an actual thesis with this very title, albeit defended by Stephen Stanley in 1998 at the University of Birmingham. that is, Birmingham, UK [near Coventry]. Apart from this amusing trivia piece, I also enjoyed watching the first episodes of the series, the main actor being really outstanding in her acting, and the plot unfolding rather nicely, except for the chess games that are unrealistically hurried, presumably because watching people thinking is anathema on TV! The representation of misogyny at the time is however most realistic (I presume|!) and definitely shocking. (The first competition game when Beth Hamon loses is somewhat disappointing as failing to predict a Queen exchange is implausible at this level…) However, the growing self-destructive behaviour of Beth made me cringe to the point of stopping the series. The early episodes also reminded me of the days when my son had started playing chess with me, winning on a regular basis, had then joined a Saturday chess nearby, was moved to the adult section within a few weeks, and … stopped altogether a few weeks later as he (mistakenly) thought the older players were making fun of him!!! He never got to any competitive level but still plays on a regular basis and trashes me just as regularly. Coincidence or not, the Guardian has a “scandalous” chess story to relate last week,  when the Dutch champion defeated the world top two players, with one game won by him having prepared the Najdorf Sicilian opening up to the 17th round! (The chess problem below is from the same article but relates to Antonio Medina v Svetozar Gligoric, Palma 1968.)