## Archive for probability theory

## a random day, in Paris

Posted in Statistics, University life with tags conference, IHP, Institut Henri Poincaré, journée aléatoire, probability theory, SFDS, SMAI, SMF, statistical inference, Statistics on September 28, 2022 by xi'an## In Bayesian statistics, data is considered nonrandom…

Posted in Books, Statistics, University life with tags Bayesian foundations, conditional probability, cross validated, probability theory, randomness, The Bayesian Choice on July 12, 2021 by xi'an**A** rather weird question popped up on X validated, namely why does Bayesian analysis rely on a sampling distribution if the data is nonrandom. While a given sample is is indeed a deterministic object and hence *nonrandom* from this perspective!, I replied that on the opposite Bayesian analysis was setting the observed data as the realisation of a random variable in order to condition upon this realisation to construct a posterior distribution on the parameter. Which is quite different from calling it *nonrandom*! But, presumably putting too much meaning and spending too much time on this query, I remain somewhat bemused by what line of thought led to this question…

## essentials of probability theory for statisticians

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags asymptotics, book review, central limit theorem, CHANCE, Cydonia, face on Mars, Glivenko-Cantelli Theorem, Henri Lebesgue, Lebesque integration, measure theory, pareidolia, probability theory, quincunx on April 25, 2020 by xi'an**O**n 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.]*

## Mea Culpa

Posted in Statistics with tags Bayesian Analysis, Bayesian foundations, book review, E.T. Jaynes, improper posteriors, improper prior, information, John Skilling, likelihood function, marginalisation paradoxes, prior information, probability theory on April 10, 2020 by xi'an*[A quote from Jaynes about improper priors that I had missed in his book, Probability Theory.]*

For many years, the present writer was caught in this error just as badly as anybody else, because Bayesian calculations with improper priors continued to give just the reasonable and clearly correct results that common sense demanded. So warnings about improper priors went unheeded; just that psychological phenomenon. Finally, it was the marginalization paradox that forced recognition that we had only been lucky in our choice of problems. If we wish to consider an improper prior, the only correct way of doing it is to approach it as a well-defined limit of a sequence of proper priors. If the correct limiting procedure should yield an improper posterior pdf for some parameter α, then probability theory is telling us that the prior information and data are too meager to permit any inferences about α. Then the only remedy is to seek more data or more prior information; probability theory does not guarantee in advance that it will lead us to a useful answer to every conceivable question.Generally, the posterior pdf is better behaved than the prior because of the extra information in the likelihood function, and the correct limiting procedure yields a useful posterior pdf that is analytically simpler than any from a proper prior. The most universally useful results of Bayesian analysis obtained in the past are of this type, because they tended to be rather simple problems, in which the data were indeed so much more informative than the prior information that an improper prior gave a reasonable approximation – good enough for all practical purposes – to the strictly correct results (the two results agreed typically to six or more significant figures).

In the future, however, we cannot expect this to continue because the field is turning to more complex problems in which the prior information is essential and the solution is found by computer. In these cases it would be quite wrong to think of passing to an improper prior. That would lead usually to computer crashes; and, even if a crash is avoided, the conclusions would still be, almost always, quantitatively wrong. But, since likelihood functions are bounded, the analytical solution with proper priors is always guaranteed to converge properly to finite results; therefore it is always possible to write a computer program in such a way (avoid underflow, etc.) that it cannot crash when given proper priors. So, even if the criticisms of improper priors on grounds of marginalization were unjustified,it remains true that in the future we shall be concerned necessarily with proper priors.