Le Monde puzzle [#1083]

Posted in Books, Kids, R, Travel with tags , , , , , , on February 7, 2019 by xi'an

A Le Monde mathematical puzzle that seems hard to solve without the backup of a computer (and just simple enough to code on a flight to Montpellier):

Given the number N=2,019, find a decomposition of N as a sum of non-trivial powers of integers such that (a) the number of integers in the sum is maximal or (b) all powers are equal to 4.  Is it possible to write N as a sum of two powers?

It is straightforward to identify all possible terms in these sums by listing all powers of integers less than N

```pool=(1:trunc(sqrt(2019)))^2
for (pow in 3:11)
pool=unique(c(pool,(2:trunc(2019^(1/pow)))^pow))
```

which leads to 57 distinct powers. Sampling at random from this collection at random produces a sum of 21 perfect powers:

` 1+4+8+9+16+25+27+32+36+49+64+81+100+121+125+128+144+169+196+243+441`

But looking at the 22 smallest numbers in the pool of powers leads to 2019, which is a sure answer. Restricting the terms to powers of 4 leads to the sequence

1⁴+2⁴+3⁴+5⁴+6⁴ = 2019

And starting from the pools of all possible powers in a decomposition of 2019 as the sum of two powers shows this is impossible.

MASH in Le Monde

Posted in Statistics with tags , , , , , , , , on January 25, 2019 by xi'an

exciting week[s]

Posted in Mountains, pictures, Running, Statistics with tags , , , , , , , , , , , , , , on June 27, 2017 by xi'an

The past week was quite exciting, despite the heat wave that hit Paris and kept me from sleeping and running! First, I made a two-day visit to Jean-Michel Marin in Montpellier, where we discussed the potential Peer Community In Computational Statistics (PCI Comput Stats) with the people behind PCI Evol Biol at INRA, Hopefully taking shape in the coming months! And went one evening through a few vineyards in Saint Christol with Jean-Michel and Arnaud. Including a long chat with the owner of Domaine Coste Moynier. [Whose domain includes the above parcel with views of Pic Saint-Loup.] And last but not least! some work planning about approximate MCMC.

On top of this, we submitted our paper on ABC with Wasserstein distances [to be arXived in an extended version in the coming weeks], our revised paper on ABC consistency thanks to highly constructive and comments from the editorial board, which induced a much improved version in my opinion, and we received a very positive return from JCGS for our paper on weak priors for mixtures! Next week should be exciting as well, with BNP 11 taking place in downtown Paris, at École Normale!!!

Grothendieck’s papers on-line!

Posted in Books, Kids, University life with tags , , , on May 10, 2017 by xi'an

Today, the University of Montpellier will put on-line the series of 18,000 pages of manuscripts of Alexandre Groethendieck that it had digitised a few years ago. Thanks to the efforts of Jean-Michel Marin, the nearly incomprehensible legal imbroglio on the rights of both the University and the children of Groethendieck has been unravelled, meaning that the University is now allowed to make the manuscripts available, while the children have the sole property of the dozens of thousands of pages written by Groethendieck till his death. It is hard to imagine how such a volume can be efficiently explored and exploited to uncover new mathematical advances made by Groethendieck in the last and secluded part of his life, but at last the raw material is available for all to try.

SPA 2015 Oxford

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on July 14, 2015 by xi'an

Today I gave a talk on Approximate Bayesian model choice via random forests at the yearly SPA (Stochastic Processes and their Applications) 2015 conference, taking place in Oxford (a nice town near Warwick) this year. In Keble College more precisely. The slides are below and while they are mostly repetitions of earlier slides, there is a not inconsequential novelty in the presentation, namely that I included our most recent and current perspective on ABC model choice. Indeed, when travelling to Montpellier two weeks ago, we realised that there was a way to solve our posterior probability conundrum!

Despite the heat wave that rolled all over France that week, we indeed figured out a way to estimate the posterior probability of the selected (MAP) model, way that we had deemed beyond our reach in previous versions of the talk and of the paper. The fact that we could not provide an estimate of this posterior probability and had to rely instead on a posterior expected loss was one of the arguments used by the PNAS reviewers in rejecting the paper. While the posterior expected loss remains a quantity worth approximating and reporting, the idea that stemmed from meeting together in Montpellier is that (i) the posterior probability of the MAP is actually related to another posterior loss, when conditioning on the observed summary statistics and (ii) this loss can be itself estimated via a random forest, since it is another function of the summary statistics. A posteriori, this sounds trivial but we had to have a new look at the problem to realise that using ABC samples was not the only way to produce an estimate of the posterior probability! (We are now working on the revision of the paper for resubmission within a few week… Hopefully before JSM!)