Archive for Riemann

Winter workshop, Gainesville (day 2)

Posted in pictures, Running, Travel, University life, Wines with tags , , , , , , , , , , , , , on January 21, 2013 by xi'an

view from the Emerson Alumni Hall auditorium,  UF, Gainesville, Florida, Jan. 19, 2013On day #2, besides my talk on “empirical Bayes” (ABCel) computation (mostly recycled from Varanasi, photos included), Christophe Andrieu gave a talk on exact approximations, using unbiased estimators of the likelihood and characterising estimators garanteeing geometric convergence (bounded weights, essentially, which is a condition popping out again and again in the Monte Carlo literature). Then Art Owen (father of empirical likelihood among other things!) spoke about QMC for MCMC, a topic that always intringued me.

Indeed, while I see the point of using QMC for specific integration problems, I am more uncertain about its relevance for statistics as a simulation device. Having points uniformly distributed over the unit hypercube in a much more efficient way than a random sample is not helping much when only a tiny region of the unit hypercube, namely the one where the likelihood concentrates, matters. (In other words, we are rarely interested in the uniform distribution over the unit hypercube: we instead want to simulate from a highly irregular and definitely concentrated distribution.) I have the same reservation about the applicability of stratified sampling: the strata have to be constructed in relation with the target distribution. The method Art advocates using a CUD (completely uniformly distributed) sequence as the underlying (deterministic) pseudo-unifom sequence. Highly interesting and I want to read the paper in greater details, but the fact that most simulation steps use a random number of uniforms seems detrimental to the performances of the method in general.

alligator in Lake Alice, Gainesville, Florida, Jan. 19, 2013After a lunch break at a terrific BBQ place, with a stop at Lake Alice to watch the alligator(s) I had missed during my morning run, I was able this time to attend till the end Xiao-Li Meng’s talk, where he presented new improvements on bridge sampling based on location-scale (or warping) transforms of the original two-samples to make them share mean and variance. Hani Doss concluded the meeting with a talk on the computation of Bayes factors when using (non-parametric) Dirichlet mixture priors, whose resolution does not require simulations for each value of the scale parameter of the Dirichlet prior, thanks to a Radon-Nykodim derivative representation. (Which nicely connected with Art’s talk in that the latter mentioned therein that most simulation methods are actually based on Riemann integration rather than Lebesgue integration. Hani’s representation is not, with nested sampling being another example.)

We ended up the day with a(nother) barbecue outside, under the stars, in the peace and quiet of a local wood, with wine and laughs, just like George would have concluded the workshop. This was a fitting ending to a meeting dedicated to his memory…

the universe in zero words

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , on May 30, 2012 by xi'an

The universe in zero words: The story of mathematics as told through equations is a book with a very nice cover: in case you cannot see the details on the picture, what looks like stars on a bright night sky are actually equations discussed in the book (plus actual stars!)…

The universe in zero words is written by Dana Mackenzie (check his website!) and published by Princeton University Press. (I received it in the mail from John Wiley for review, prior to its publication on May 16, nice!) It reads well and quick: I took it with me in the métro one morning and was half-way through it the same evening, as the universe in zero words remains on the light side, esp. for readers with a high-school training in math. The book strongly reminded me (at times) of my high school years and of my fascination for Cardano’s formula and the non-Euclidean geometries. I was also reminded of studying quaternions for a short while as an undergraduate by the (arguably superfluous) chapter on Hamilton. So a pleasant if unsurprising read, with a writing style that is not always at its best, esp. after reading Bill Bryson’s “Seeing Further: The Story of Science, Discovery, and the Genius of the Royal Society“, and a book unlikely to bring major epiphanies to the mathematically inclined. If well-documented, free of typos, and engaging into some mathematical details (accepting to go against the folk rule that “For every equation you put in, you will lose half of your audience.” already mentioned in Diaconis and Graham’s book). With alas a fundamental omission: no trace is found therein of Bayes’ formula! (The very opposite of Bryson’s introduction, who could have arguably stayed away from it.) The closest connection with statistics is the final chapter on the Black-Scholes equation, which does not say much about probability…. It is of course the major difficulty with the exercise of picking 24 equations out of the history of maths and physics that some major and influential equations had to be set aside… Maybe the error was in covering (or trying to cover) formulas from physics as well as from maths. Now, rather paradoxically (?) I learned more from the physics chapters: for instance, the chapters on Maxwell’s, Einstein’s, and Dirac’s formulae are very well done. The chapter on the fundamental theorem of calculus is also appreciable.

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