another drawer of socks

A socks riddle from the Riddler but with no clear ABC connection! Twenty-eight socks from fourteen pairs of socks are taken from a drawer, one by one, and laid on a surface that only fit nine socks at a time, with complete pairs removed. What is the probability that all pairs are stored without running out of space? No orphan socks then!!

Writing an R code for this experiment is straightforward

for(v in 1:1e6){
 S=sample(rep(1:14,2))
 x=S[1]
 for(t in 2:18){
  if(S[t]%in%x){x=x[S[t]!=x]}else{x=c(x,S[t])}
  if(sum(!!x)>9){
    F=F+1;break()}}}

and it returns a value quite close to 0.7 for the probability of success. I was expecting a less brute-force resolution but the the Riddler only provided the answer of 70.049 based on the above tree of probabilities (which I was too lazy to code).

a glaringly long explanation

It is funny that, when I am teaching the rudiments of Bayesian statistics to my undergraduate students in Paris-Dauphine, including ABC via Rasmus’ socks, specific questions about the book (The Bayesian Choice) start popping up on X validated! Last week was about the proof that ABC is exact when the tolerance is zero. And the summary statistic sufficient.

This week is about conjugate distributions for exponential families (not that there are many others!). Which led me to explain both the validation of the conjugacy and the derivation of the posterior expectation of the mean of the natural sufficient statistic in far more details than in the book itself. Hopefully in a profitable way.

Astrostatistics school

What a wonderful week at the Astrostat [Indian] summer school in Autrans! The setting was superb, on the high Vercors plateau overlooking both Grenoble [north] and Valence [west], with the colours of the Fall at their brightest on the foliage of the forests rising on both sides of the valley and a perfect green on the fields at the centre, with sun all along, sharp mornings and warm afternoons worthy of a late Indian summer, too many running trails [turning into X country ski trails in the Winter] to contemplate for a single week [even with three hours of running over two days], many climbing sites on the numerous chalk cliffs all around [but a single afternoon for that, more later in another post!]. And of course a group of participants eager to learn about Bayesian methodology and computational algorithms, from diverse [astronomy, cosmology and more] backgrounds, trainings and countries. I was surprised at the dedication of the participants travelling all the way from Chile, Péru, and Hong Kong for the sole purpose of attending the school. David van Dyk gave the first part of the school on Bayesian concepts and MCMC methods, Roberto Trotta the second part on Bayesian model choice and hierarchical models, and myself a third part on, surprise, surprise!, approximate Bayesian computation. Plus practicals on R.

As it happens Roberto had to cancel his participation and I turned for a session into Christian Roberto, presenting his slides in the most objective possible fashion!, as a significant part covered nested sampling and Savage-Dickey ratios, not exactly my favourites for estimating constants. David joked that he was considering postponing his flight to see me talk about these, but I hope I refrained from engaging into controversy and criticisms… If anything because this was not of interest for the participants. Indeed when I started presenting ABC through what I thought was a pedestrian example, namely Rasmus Baath’s socks, I found that the main concern was not running an MCMC sampler or a substitute ABC algorithm but rather an healthy questioning of the construction of the informative prior in that artificial setting, which made me quite glad I had planned to cover this example rather than an advanced model [as, e.g., one of those covered in the packages abc, abctools, or abcrf]. Because it generated those questions about the prior [why a Negative Binomial? why these hyperparameters? &tc.] and showed how programming ABC turned into a difficult exercise even in this toy setting. And while I wanted to give my usual warning about ABC model choice and argue for random forests as a summary selection tool, I feel I should have focussed instead on another example, as this exercise brings out so clearly the conceptual difficulties with what is taught. Making me quite sorry I had to leave one day earlier. [As did missing an extra run!] Coming back by train through the sunny and grape-covered slopes of Burgundy hills was an extra reward [and no one in the train commented about the local cheese travelling in my bag!]

 

Rasmus’ socks fit perfectly!

Following the previous post on Rasmus’ socks, I took the opportunity of a survey on ABC I am currently completing to compare the outcome of his R code with my analytical derivation. After one quick correction [by Rasmus] of a wrong representation of the Negative Binomial mean-variance parametrisation [by me], I achieved this nice fit…