a football post?!

I am not interested in football, neither as a player (a primary school trauma when I was the last being picked!) or as a fan, contrary to my dad (who was a football referee in his youth) and my kids, but Gareth Roberts (University of Warwick) and Jeff Rosenthal wrote a paper on football draws for the (FIFA) World Cup, infamously playing in Qatar by the end of the year, which Gareth presented in a Warwick seminar.

For this tournament, there are 32 teams, first playing against opponent teams supposedly drawn from a uniform distribution over all draw assignments, within 8 groups of 4 teams, with constraints like 1-2 EU teams per group, 0-1 from the other regions. As done at the moment and on TV, the tournament is filled one team at time by drawing from Pot 1, then Pot 2, then Pot 3, & Pot 4. &tc.. Applying the constraints one draw at a time, conditional on the past draws and the constraints, rather obviously creates non-uniformity! Uniformity would be achievable by rejection sampling (with a success probability of 1/540!) But this is not televisesque enough…

A debiasing solution is found by using several balls for each team in the right proportion, correcting for the sequential draws. Still impractical when requiring 10¹⁴ balls…!

The fun in their paper is that the problem can be formulated as a particle filter, estimating the right probabilities by randomising the number of balls [hidden randomness] and estimating the probability for team j to be included by a few thousands draws. With some stratified sampling on the side to minimise randomness. Removing the need for the (intractable?) distribution is thus achieved by retrospective sampling, as in pseudo-marginal MCMC. Alternatively, one could swap pairs of teams by a simplistic MCMC algorithm, with no worry about stationarity and the possibility of on-screen draws. (Jeff devised a Java applet to simulate an actual draw.) Obviously, it is still a far stretch that this proposal will be implemented for the next World Cup. If so, I will watch it!

set-valued sufficient statistic

While the classical definition of a statistic is one of a real valued random variable or vector, less usual situations call for broader definitions… For instance, in an homework problem from Mark Schervish’s Theory of Statistics, a sample from the uniform distribution of a ball of unknown centre θ and radius ς is associated with the convex hull of said sample as “sufficient statistic”, albeit the object being a set. Similarly, if the radius ς is known, the set made of the intersection of all the balls of radius ς centred at the observations is sufficient, in that the likelihood is constant for θ inside and zero outside. As discussed in this X validated question, this does not define an optimal estimator of the center θ, while Pitman’s best location equivariant does, while the centre of this sufficient set, but it is not sufficient as a statistic and is not necessarily the MVUE, if unbiased.

how can a posterior be uniform?

A bemusing question from X validated:

How can we have a posterior distribution that is a uniform distribution?

With the underlying message that a uniform distribution does not depend on the data, since it is uniform! While it is always possible to pick the parameterisation a posteriori so that the posterior is uniform, by simply using the inverse cdf transform, or to pick the prior a posteriori so that the prior cancels the likelihood function, there exist more authentic discrete examples of a data realisation leading to a uniform distribution, as eg in the Multinomial model. I deem the confusion to stem from the impression either that uniform means non-informative (what we could dub Laplace’s daemon!) or that it could remain uniform for all realisations of the sampled rv.

sampling the mean

A challenge found on the board of the coffee room at CEREMADE, Université Paris Dauphine:

When sampling with replacement three numbers in {0,1,…,N}, what is the probability that their average is (at least) one of the three?

With a (code-golfed!) brute force solution of

mean(!apply((a<-matrix(sample(0:n,3e6,rep=T),3)),2,mean)-apply(a,2,median))

producing a graph pretty close to 3N/2(N+1)² (which coincides with a back-of-the-envelope computation):

45 votes for Jensen’s inequality

Following a question on X validated as to why the mean of the log of a uniform distribution is not log(0.5), I replied with the obvious link to Jensen’s inequality and the more general if equally obvious remark that expectation was rarely invariant under transforms and ended up with an high number of up-votes on that answer. Which bemuses me given the basic question and equally basic answer..!