Archive for Lebesque integration

essentials of probability theory for statisticians

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on April 25, 2020 by xi'an

On yet another confined sunny lazy Sunday morning, I read through Proschan and Shaw’s Essentials of Probability Theory for Statisticians, a CRC Press book that was sent to me quite a while ago for review. The book was indeed published in 2016. Before moving to serious things, let me evacuate the customary issue with the cover. I have trouble getting the point of the “face on Mars” being adopted as the cover of a book on probability theory (rather than a book on, say, pareidolia). There is a brief paragraph on post-facto probability calculations, stating how meaningless the question of the probability of this shade appearing on a Viking Orbiter picture by “chance”, but this is so marginal I would have preferred any other figure from the book!

The book plans to cover the probability essentials for dealing with graduate level statistics and in particular convergence, conditioning, and paradoxes following from using non-rigorous approaches to probability. A range that completely fits my own prerequisite for statistics students in my classes and that of course involves the recourse to (Lebesgue) measure theory. And a goal that I find both commendable and comforting as my past experience with exchange students led me to the feeling that rigorous probability theory was mostly scrapped from graduate programs. While the book is not extremely formal, it provides a proper motivation for the essential need of measure theory to handle the complexities of statistical analysis and in particular of asymptotics. It thus relies as much as possible on examples that stem from or relate to statistics, even though most examples may appear as standard to senior readers. For instance the consistency of the sample median or a weak version of the Glivenko-Cantelli theorem. The final chapter is dedicated to applications (in the probabilist’ sense!) that emerged from statistical problems. I felt these final chapters were somewhat stretched compared with what they could have been, as for instance with the multiple motivations of the conditional expectation, but this simply makes for more material. If I had to teach this material to students, I would certainly rely on the book! in particular because of the repeated appearances of the quincunx for motivating non-Normal limites. (A typo near Fatou’s lemma missed the dominating measure. And I did not notice the Riemann notation dx being extended to the measure in a formal manner.)

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE.]

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…