## a probabilistic proof to a quasi-Monte Carlo lemma

Posted in Books, Statistics, Travel, University life with tags , , , , , on November 17, 2014 by xi'an

As I was reading in the Paris métro a new textbook on Quasi-Monte Carlo methods, Introduction to Quasi-Monte Carlo Integration and Applications, written by Gunther Leobacher and Friedrich Pillichshammer, I came upon the lemma that, given two sequences on (0,1) such that, for all i’s,

$|u_i-v_i|\le\delta\quad\text{then}\quad\left|\prod_{i=1}^s u_i-\prod_{i=1}^s v_i\right|\le 1-(1-\delta)^s$

and the geometric bound made me wonder if there was an easy probabilistic proof to this inequality. Rather than the algebraic proof contained in the book. Unsurprisingly, there is one based on associating with each pair (u,v) a pair of independent events (A,B) such that, for all i’s,

$A_i\subset B_i\,,\ u_i=\mathbb{P}(A_i)\,,\ v_i=\mathbb{P}(B_i)$

and representing

$\left|\prod_{i=1}^s u_i-\prod_{i=1}^s v_i\right| = \mathbb{P}(\cap_{i=1}^s A_i) - \mathbb{P}(\cap_{i=1}^s B_i)\,.$

Obviously, there is no visible consequence to this remark, but it was a good way to switch off the métro hassle for a while! (The book is under review and the review will hopefully be posted on the ‘Og as soon as it is completed.)

## métro static

Posted in pictures, Running with tags , , , , , , , on September 27, 2014 by xi'an

“Mon premier marathon je le fais en courant.” [I will do my first marathon running.]

## métro static

Posted in Kids, Travel with tags , , on March 26, 2014 by xi'an

[heard in the métro this morning]

“…les équations à deux inconnues ça va encore, mais à trois inconnues, c’est trop dur!”

[“…systems of equations with two unknowns are still ok, but with three variables it is too hard!”]

## métro static

Posted in Kids, Travel with tags , , , on June 15, 2013 by xi'an

“…tiens, on est déjà à François Mitterand… ah non, c’est Musée d’Orsay…”

“…Musée d’Orsay, c’est pas là qu’on y est déjà allé une fois?”

“…oui, oui, là où y a des statues en cire.”

“…mais non, ça, c’est Musée Grévin…”

Later, on that very same trip, I tried to help an old lady who had to change lines twice. But she was so confused in her head that she did not recognise me when I told her this was her first switch. She looked through me as if she had never seen me. And presumably she had not. I hope the poor soul eventually made it to her final destination…

## métro static

Posted in Travel with tags , , on March 25, 2013 by xi'an

“…tiens, je croyais qu’il y avait plus de morts cette année mais en fait c’est la même chose…

..d’habitude, il y en a trente et cette année c’est vingt-deux…”

## making a random walk geometrically ergodic

Posted in R, Statistics with tags , , , , , , , , , on March 2, 2013 by xi'an

While a random walk Metropolis-Hastings algorithm cannot be uniformly ergodic in a general setting (Mengersen and Tweedie, AoS, 1996), because it needs more energy to leave far away starting points, it can be geometrically ergodic depending on the target (and the proposal). In a recent Annals of Statistics paper, Leif Johnson and Charlie Geyer designed a trick to turn a random walk Metropolis-Hastings algorithm into a geometrically ergodic random walk Metropolis-Hastings algorithm by virtue of an isotropic transform (under the provision that the original target density has a moment generating function). This theoretical result is complemented by an R package called mcmc. (I have not tested it so far, having read the paper in the métro.) The examples included in the paper are however fairly academic and I wonder how the method performs in practice, on truly complex models, in particular because the change of variables relies on (a) an origin and (b) changing the curvature of space uniformly in all dimensions. Nonetheless, the idea is attractive and reminds me of a project of ours with Randal Douc,  started thanks to the ‘Og and still under completion.

## scary…

Posted in Kids, Travel with tags , , on February 15, 2013 by xi'an

(Announcement heard in my métro station this morning…)

Un petit garçon de 4 ans est tout seul dans la gare. Si vous l’apercevez, merci de l’accompagner jusqu’au guichet.

(The little boy must have been found right away as there was no mention later of a missing child in the news. This reminded me of the only time and of the terribly long seconds when we thought our four year old son had vanished, when he was simply playing a terribly long hide-and-seek…)