Archive for measure theory

Bertrand’s tartine

Posted in Books, Kids, pictures, Statistics with tags , , , , , , , , , , on November 25, 2022 by xi'an

A riddle from The Riddler on cutting a square (toast) into two parts and keeping at least 25% of the surface on each part while avoiding Bertrand’s paradox. By defining the random cut as generated by two uniform draws over the periphery of the square. Meaning that ¼ of the draws are on the same side, ½ on adjacent sides and again ¼ on opposite sides. Meaning one has to compute

P(UV>½)= ½(1-log(2))


P(½(U+V)∈(¼,¾))= ¾

Resulting in a probability of 0.2642 (checked by simulation)

conditioning on insufficient statistics in Bayesian regression

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on October 23, 2021 by xi'an

“…the prior distribution, the loss function, and the likelihood or sampling density (…) a healthy skepticism encourages us to question each of them”

A paper by John Lewis, Steven MacEachern, and Yoonkyung Lee has recently appeared in Bayesian Analysis. Starting with the great motivation of a misspecified model requiring the use of a (thus necessarily) insufficient statistic and moving to their central concern of simulating the posterior based on that statistic.

Model misspecification remains understudied from a B perspective and this paper is thus most welcome in addressing the issue. However, when reading through, one of my criticisms is in defining misspecification as equivalent to outliers in the sample. An outlier model is an easy case of misspecification, in the end, since the original model remains meaningful. (Why should there be “good” versus “bad” data) Furthermore, adding a non-parametric component for the unspecified part of the data would sound like a “more Bayesian” alternative. Unrelated, I also idly wondered at whether or not normalising flows could be used in this instance..

The problem in selecting a T (Darjeeling of course!) is not really discussed there, while each choice of a statistic T leads to a different signification to what misspecified means and suggests a comparison with Bayesian empirical likelihood.

“Acceptance rates of this [ABC] algorithm can be intolerably low”

Erm, this is not really the issue with ABC, is it?! Especially when the tolerance is induced by the simulations themselves.

When I reached the MCMC (Gibbs?) part of the paper, I first wondered at its relevance for the mispecification issues before realising it had become the focus of the paper. Now, simulating the observations conditional on a value of the summary statistic T is a true challenge. I remember for instance George Casella mentioning it in association with a Student’s t sample in the 1990’s and Kerrie and I having an unsuccessful attempt at it in the same period. Persi Diaconis has written several papers on the problem and I am thus surprised at the dearth of references here, like the rather recent Byrne and Girolami (2013), Florens and Simoni (2015), or Bornn et al. (2019). In the present case, the  linear model assumed as the true model has the exceptional feature that it leads to a feasible transform of an unconstrained simulation into a simulation with fixed statistics, with no measure theoretic worries if not free from considerable efforts to establish the operation is truly valid… And, while simulating (θ,y) makes perfect sense in an insufficient setting, the cost is then precisely the same as when running a vanilla ABC. Which brings us to the natural comparison with ABC. While taking ε=0 may sound as optimal for being “exact”, it is not from an ABC perspective since the convergence rate of the (summary) statistic should be roughly the one of the tolerance (Fearnhead and Liu, Frazier et al., 2018).

“[The Borel Paradox] shows that the concept of a conditional probability with regard to an isolated given hypothesis whose probability equals 0 is inadmissible.” A. Колмого́ров (1933)

As a side note for measure-theoretic purists, the derivation of the conditional of y given T(y)=T⁰ is arbitrary since the event has probability zero (ie, the conditioning set is of measure zero). See the Borel-Kolmogorov paradox. The computations in the paper are undoubtedly correct, but this is only one arbitrary choice of a transform (or conditioning σ-algebra).

conditioning an algorithm

Posted in Statistics with tags , , , , , , , , , , , on June 25, 2021 by xi'an

A question of interest on X validated: given a (possibly black-box) algorithm simulating from a joint distribution with density [wrt a continuous measure] p(z,y) (how) is it possible to simulate from the conditional p(y|z⁰)? Which reminded me of a recent paper by Lindqvist et al. on conditional Monte Carlo. Which zooms on the simulation of a sample X given the value of a sufficient statistic, T(X)=t, revolving about pivotal quantities and inversions à la fiducial statistics, following an earlier Biometrika paper by Lindqvist & Taraldsen, in 2005. The idea is to write

X=\chi(U,\theta)\qquad T(X)=\tau(U,\theta)

where U has a distribution that depends on θ, to solve τ(u,θ)=t in θ for a given pair (u,t) with solution θ(u,t) and to generate u conditional on this solution. But this requires getting “under the hood” of the algorithm to such an extent as not answering the original question, or being open to other solutions using the expression for the joint density p(z,y)… In a purely black box situation, ABC appears as the natural if approximate solution.

principles of uncertainty (second edition)

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , on July 21, 2020 by xi'an

A new edition of Principles of Uncertainty is about to appear. I was asked by CRC Press to review the new book and here are some (raw) extracts from my review. (Some comments may not apply to the final and published version, mind.)

In Chapter 6, the proof of the Central Limit Theorem utilises the “smudge” technique, which is to add an independent noise to both the sequence of rvs and its limit. This is most effective and reminds me of quite a similar proof Jacques Neveu used in its probability notes in Polytechnique. Which went under the more formal denomination of convolution, with the same (commendable) purpose of avoiding Fourier transforms. If anything, I would have favoured a slightly more condensed presentation in less than 8 pages. Is Corollary 6.5.8 useful or even correct??? I do not think so because the non-centred average rescaled by √n diverges almost surely. For the same reason, I object to the very first sentence of Section 6.5 (p.246)

In Chapter 7, I found a nice mention of (Hermann) Rubin’s insistence on not separating probability and utility as only the product matters. And another fascinating quote from Keynes, not from his early statistician’s years, but in 1937 as an established economist

“The sense in which I am using the term uncertain is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to the summed.”

(is the last sentence correct? I would have expected, pardon my French!, “to be summed”). Further interesting trivia on the criticisms of utility theory, including de Finetti’s role and his own lack of connection with subjective probability principles.

In Chapter 8, a major remark (iii) is found p.293 about the fact that a conjugate family requires a dominating measure (although this is expressed differently since the book shies away from introducing measure theory, ) reminds me of a conversation I had with Jay when I visited Carnegie Mellon in 2013 (?). Which exposes the futility of seeing conjugate priors as default priors. It is somewhat surprising that a notion like admissibility appears as a side quantity when discussing Stein’s paradox in 8.2.1 [and then later in Section 9.1.3] while it seems to me to be central to Bayesian decision theory, much more than the epiphenomenon that Stein’s paradox represents in the big picture. But the book dismisses minimaxity even faster in Section 9.1.4:

As many who suffer from paranoia have discovered, one can always dream-up an even worse possibility to guard against. Thus, the minimax framework is unstable. (p.336)

Interesting introduction of the Wishart distribution to kindly handle random matrices and matrix Jacobians, with the original space being the p(p+1)/2 real space (implicitly endowed with the Lebesgue measure). Rather than a more structured matricial space. A font error makes Corollary 8.7.2 abort abruptly. The space of positive definite matrices is mentioned in Section8.7.5 but still (implicitly) corresponds to the common p(p+1)/2 real Euclidean space. Another typo in Theorem 8.9.2 with a Frenchised version of Dirichlet, Dirichelet. Followed by a Dirchlet at the end of the proof (p.322). Again and again on p.324 and on following pages. I would object to the singular in the title of Section 8.10 as there are exponential families rather than a single one. With no mention made of Pitman-Koopman lemma and its consequences, namely that the existence of conjugacy remains an epiphenomenon. Hence making the amount of pages dedicated to gamma, Dirichlet and Wishart distributions somewhat excessive.

In Chapter 9, I noticed (p.334) a Scheffe that should be Scheffé (and again at least on p.444). (I love it that Jay also uses my favorite admissible (non-)estimator, namely the constant value estimator with value 3.) I wonder at the worth of a ten line section like 9.3, when there are delicate issues in handling meta-analysis, even in a Bayesian mood (or mode). In the model uncertainty section, Jay discuss the (im)pertinence of both selecting one of the models and setting independent priors on their respective parameters, with which I disagree on both levels. Although this is followed by a more reasonable (!) perspective on utility. Nice to see a section on causation, although I would have welcomed an insert on the recent and somewhat outrageous stand of Pearl (and MacKenzie) on statisticians missing the point on causation and counterfactuals by miles. Nonparametric Bayes is a new section, inspired from Ghahramani (2005). But while it mentions Gaussian and Dirichlet [invariably misspelled!] processes, I fear it comes short from enticing the reader to truly grasp the meaning of a prior on functions. Besides mentioning it exists, I am unsure of the utility of this section. This is one of the rare instances where measure theory is discussed, only to state this is beyond the scope of the book (p.349).

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.]

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