Archive for measure theory

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.]

Probability and Bayesian modeling [book review]

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , , , , , , , , , , on March 26, 2020 by xi'an

Probability and Bayesian modeling is a textbook by Jim Albert [whose reply is included at the end of this entry] and Jingchen Hu that CRC Press sent me for review in CHANCE. (The book is also freely available in bookdown format.) The level of the textbook is definitely most introductory as it dedicates its first half on probability concepts (with no measure theory involved), meaning mostly focusing on counting and finite sample space models. The second half moves to Bayesian inference(s) with a strong reliance on JAGS for the processing of more realistic models. And R vignettes for the simplest cases (where I discovered R commands I ignored, like dplyr::mutate()!).

As a preliminary warning about my biases, I am always reserved at mixing introductions to probability theory and to (Bayesian) statistics in the same book, as I feel they should be separated to avoid confusion. As for instance between histograms and densities, or between (theoretical) expectation and (empirical) mean. I therefore fail to relate to the pace and tone adopted in the book which, in my opinion, seems to dally on overly simple examples [far too often concerned with food or baseball] while skipping over the concepts and background theory. For instance, introducing the concept of subjective probability as early as page 6 is laudable but I doubt it will engage fresh readers when describing it as a measurement of one’s “belief about the truth of an event”, then stressing that “make any kind of measurement, one needs a tool like a scale or ruler”. Overall, I have no particularly focused criticisms on the probability part except for the discrete vs continuous imbalance. (With the Poisson distribution not covered in the Discrete Distributions chapter. And the “bell curve” making a weird and unrigorous appearance there.) Galton’s board (no mention found of quincunx) could have been better exploited towards the physical definition of a prior, following Steve Stiegler’s analysis, by adding a second level. Or turned into an R coding exercise. In the continuous distributions chapter, I would have seen the cdf coming first to the pdf, rather than the opposite. And disliked the notion that a Normal distribution was supported by an histogram of (marathon) running times, i.e. values lower bounded by 122 (at the moment). Or later (in Chapter 8) for Roger Federer’s serving times. Incidentally, a fun typo on p.191, at least fun for LaTeX users, as

f_{Y\ mid X}

with an extra space between `\’ and `mid’! (I also noticed several occurrences of the unvoidable “the the” typo in the last chapters.) The simulation from a bivariate Normal distribution hidden behind a customised R function sim_binom() when it could have been easily described as a two-stage hierarchy. And no comment on the fact that a sample from Y-1.5X could be directly derived from the joint sample. (Too unconscious a statistician?)

When moving to Bayesian inference, a large section is spent on very simple models like estimating a proportion or a mean, covering both discrete and continuous priors. And strongly focusing on conjugate priors despite giving warnings that they do not necessarily reflect prior information or prior belief. With some debatable recommendation for “large” prior variances as weakly informative or (worse) for Exp(1) as a reference prior for sample precision in the linear model (p.415). But also covering Bayesian model checking either via prior predictive (hence Bayes factors) or posterior predictive (with no mention of using the data twice). A very marginalia in introducing a sufficient statistic for the Normal model. In the Normal model checking section, an estimate of the posterior density of the mean is used without (apparent) explanation.

“It is interesting to note the strong negative correlation in these parameters. If one assigned informative independent priors on and , these prior beliefs would be counter to the correlation between the two parameters observed in the data.”

For the same reasons of having to cut on mathematical validation and rigour, Chapter 9 on MCMC is not explaining why MCMC algorithms are converging outside of the finite state space case. The proposal in the algorithmic representation is chosen as a Uniform one, since larger dimension problems are handled by either Gibbs or JAGS. The recommendations about running MCMC do not include how many iterations one “should” run (or other common queries on Stack eXchange), albeit they do include the sensible running multiple chains and comparing simulated predictive samples with the actual data as a  model check. However, the MCMC chapter very quickly and inevitably turns into commented JAGS code. Which I presume would require more from the students than just reading the available code. Like JAGS manual. Chapter 10 is mostly a series of examples of Bayesian hierarchical modeling, with illustrations of the shrinkage effect like the one on the book cover. Chapter 11 covers simple linear regression with some mentions of weakly informative priors,  although in a BUGS spirit of using large [enough?!] variances: “If one has little information about the location of a regression parameter, then the choice of the prior guess is not that important and one chooses a large value for the prior standard deviation . So the regression intercept and slope are each assigned a Normal prior with a mean of 0 and standard deviation equal to the large value of 100.” (p.415). Regardless of the scale of y? Standardisation is covered later in the chapter (with the use of the R function scale()) as part of constructing more informative priors, although this sounds more like data-dependent priors to me in the sense that the scale and location are summarily estimated by empirical means from the data. The above quote also strikes me as potentially confusing to the students, as it does not spell at all how to design a joint distribution on the linear regression coefficients that translate the concentration of these coefficients along y̅=β⁰+β¹x̄. Chapter 12 expands the setting to multiple regression and generalised linear models, mostly consisting of examples. It however suggests using cross-validation for model checking and then advocates DIC (deviance information criterion) as “to approximate a model’s out-of-sample predictive performance” (p.463). If only because it is covered in JAGS, the definition of the criterion being relegated to the last page of the book. Chapter 13 concludes with two case studies, the (often used) Federalist Papers analysis and a baseball career hierarchical model. Which may sound far-reaching considering the modest prerequisites the book started with.

In conclusion of this rambling [lazy Sunday] review, this is not a textbook I would have the opportunity to use in Paris-Dauphine but I can easily conceive its adoption for students with limited maths exposure. As such it offers a decent entry to the use of Bayesian modelling, supported by a specific software (JAGS), and rightly stresses the call to model checking and comparison with pseudo-observations. Provided the course is reinforced with a fair amount of computer labs and projects, the book can indeed achieve to properly introduce students to Bayesian thinking. Hopefully leading them to seek more advanced courses on the topic.

Update: Jim Albert sent me the following precisions after this review got on-line:

Thanks for your review of our recent book.  We had a particular audience in mind, specifically undergraduate American students with some calculus background who are taking their first course in probability and statistics.  The traditional approach (which I took many years ago) teaches some probability one semester and then traditional inference (focusing on unbiasedness, sampling distributions, tests and confidence intervals) in the second semester.  There didn’t appear to be any Bayesian books at that calculus-based undergraduate level and that motivated the writing of this book.  Anyway, I think your comments were certainly fair and we’ve already made some additions to our errata list based on your comments.
[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]

on confidence distributions

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , on January 10, 2018 by xi'an

As Regina Liu gave her talk at ISI this morning on fusion learning and confidence distributions, this led me to think anew about this strange notion of confidence distributions, building a distribution on the parameter space without a prior to go with it, implicitly or explicitly, and vaguely differing from fiducial inference. (As an aside, the Wikipedia page on confidence distributions is rather heavily supporting the concept and was primarily written by someone from Rutgers, where the modern version was developed. [And as an aside inside the aside, Schweder and Hjort’s book is sitting in my office, waiting for me!])

Recall that a confidence distribution is a sample dependent distribution on the parameter space, which is uniform U(0,1) [in the sample] at the “true” value of the parameter. Used thereafter as a posterior distribution. (Again, almost always without a prior to go with it. Which is an incoherence from a probabilistic perspective. not mentioning the issue of operating without a pre-defined dominating measure. This measure issue is truly bothering me!) This seems to include fiducial distributions based on a pivot, unless I am confused. As noted in the review by Nadarajah et al. Moreover, the concept of creating a pseudo-posterior out of an existing (frequentist) confidence interval procedure to create a new (frequentist) procedure does not carry an additional validation per se, as it clearly depends on the choice of the initialising procedure. (Not even mentioning the lack of invariance and the intricacy of multidimensional extensions.)

the decline of the French [maths] empire

Posted in Kids, University life with tags , , , , , , , on December 16, 2017 by xi'an

In Le Monde edition of Nov 5, an article on the difficulty of maths departments to attract students, especially in master programs and in the training of secondary school maths teachers (Agrégation & CAPES), where the number of candidates usually does not reach the number of potential positions… And also on the deep changes in the training of secondary school pupils, who over the past five years have lost a considerable amount of maths bases and hence are found missing when entering the university level. (Or, put otherwise, have a lower level in maths that implies a strong modification of our own programs and possibly the addition of an extra year or at least semester to the bachelor degree…) For instance, a few weeks ago, I realised for instance that my third year class had little idea of a conditional density and teaching measure theory at this level becomes more and more of a challenge!

10 great ideas about chance [book preview]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on November 13, 2017 by xi'an

[As I happened to be a reviewer of this book by Persi Diaconis and Brian Skyrms, I had the opportunity (and privilege!) to go through its earlier version. Here are the [edited] comments I sent back to PUP and the authors about this earlier version. All in  all, a terrific book!!!]

The historical introduction (“measurement”) of this book is most interesting, especially its analogy of chance with length. I would have appreciated a connection earlier than Cardano, like some of the Greek philosophers even though I gladly discovered there that Cardano was not only responsible for the closed form solutions to the third degree equation. I would also have liked to see more comments on the vexing issue of equiprobability: we all spend (if not waste) hours in the classroom explaining to (or arguing with) students why their solution is not correct. And they sometimes never get it! [And we sometimes get it wrong as well..!] Why is such a simple concept so hard to explicit? In short, but this is nothing but a personal choice, I would have made the chapter more conceptual and less chronologically historical.

“Coherence is again a question of consistent evaluations of a betting arrangement that can be implemented in alternative ways.” (p.46)

The second chapter, about Frank Ramsey, is interesting, if only because it puts this “man of genius” back under the spotlight when he has all but been forgotten. (At least in my circles.) And for joining probability and utility together. And for postulating that probability can be derived from expectations rather than the opposite. Even though betting or gambling has a (negative) stigma in many cultures. At least gambling for money, since most of our actions involve some degree of betting. But not in a rational or reasoned manner. (Of course, this is not a mathematical but rather a psychological objection.) Further, the justification through betting is somewhat tautological in that it assumes probabilities are true probabilities from the start. For instance, the Dutch book example on p.39 produces a gain of .2 only if the probabilities are correct.

> gain=rep(0,1e4)
> for (t in 1:1e4){
+ p=rexp(3);p=p/sum(p)
+ gain[t]=(p[1]*(1-.6)+p[2]*(1-.2)+p[3]*(.9-1))/sum(p)}
> hist(gain)

As I made it clear at the BFF4 conference last Spring, I now realise I have never really adhered to the Dutch book argument. This may be why I find the chapter somewhat unbalanced with not enough written on utilities and too much on Dutch books.

“The force of accumulating evidence made it less and less plausible to hold that subjective probability is, in general, approximate psychology.” (p.55)

A chapter on “psychology” may come as a surprise, but I feel a posteriori that it is appropriate. Most of it is about the Allais paradox. Plus entries on Ellesberg’s distinction between risk and uncertainty, with only the former being quantifiable by “objective” probabilities. And on Tversky’s and Kahneman’s distinction between heuristics, and the framing effect, i.e., how the way propositions are expressed impacts the choice of decision makers. However, it is leaving me unclear about the conclusion that the fact that people behave irrationally should not prevent a reliance on utility theory. Unclear because when taking actions involving other actors their potentially irrational choices should also be taken into account. (This is mostly nitpicking.)

“This is Bernoulli’s swindle. Try to make it precise and it falls apart. The conditional probabilities go in different directions, the desired intervals are of different quantities, and the desired probabilities are different probabilities.” (p.66)

The next chapter (“frequency”) is about Bernoulli’s Law of Large numbers and the stabilisation of frequencies, with von Mises making it the basis of his approach to probability. And Birkhoff’s extension which is capital for the development of stochastic processes. And later for MCMC. I like the notions of “disreputable twin” (p.63) and “Bernoulli’s swindle” about the idea that “chance is frequency”. The authors call the identification of probabilities as limits of frequencies Bernoulli‘s swindle, because it cannot handle zero probability events. With a nice link with the testing fallacy of equating rejection of the null with acceptance of the alternative. And an interesting description as to how Venn perceived the fallacy but could not overcome it: “If Venn’s theory appears to be full of holes, it is to his credit that he saw them himself.” The description of von Mises’ Kollectiven [and the welcome intervention of Abraham Wald] clarifies my previous and partial understanding of the notion, although I am unsure it is that clear for all potential readers. I also appreciate the connection with the very notion of randomness which has not yet found I fear a satisfactory definition. This chapter asks more (interesting) questions than it brings answers (to those or others). But enough, this is a brilliant chapter!

“…a random variable, the notion that Kac found mysterious in early expositions of probability theory.” (p.87)

Chapter 5 (“mathematics”) is very important [from my perspective] in that it justifies the necessity to associate measure theory with probability if one wishes to evolve further than urns and dices. To entitle Kolmogorov to posit his axioms of probability. And to define properly conditional probabilities as random variables (as my third students fail to realise). I enjoyed very much reading this chapter, but it may prove difficult to read for readers with no or little background in measure (although some advanced mathematical details have vanished from the published version). Still, this chapter constitutes a strong argument for preserving measure theory courses in graduate programs. As an aside, I find it amazing that mathematicians (even Kac!) had not at first realised the connection between measure theory and probability (p.84), but maybe not so amazing given the difficulty many still have with the notion of conditional probability. (Now, I would have liked to see some description of Borel’s paradox when it is mentioned (p.89).

“Nothing hangs on a flat prior (…) Nothing hangs on a unique quantification of ignorance.” (p.115)

The following chapter (“inverse inference”) is about Thomas Bayes and his posthumous theorem, with an introduction setting the theorem at the centre of the Hume-Price-Bayes triangle. (It is nice that the authors include a picture of the original version of the essay, as the initial title is much more explicit than the published version!) A short coverage, in tune with the fact that Bayes only contributed a twenty-plus paper to the field. And to be logically followed by a second part [formerly another chapter] on Pierre-Simon Laplace, both parts focussing on the selection of prior distributions on the probability of a Binomial (coin tossing) distribution. Emerging into a discussion of the position of statistics within or even outside mathematics. (And the assertion that Fisher was the Einstein of Statistics on p.120 may be disputed by many readers!)

“So it is perfectly legitimate to use Bayes’ mathematics even if we believe that chance does not exist.” (p.124)

The seventh chapter is about Bruno de Finetti with his astounding representation of exchangeable sequences as being mixtures of iid sequences. Defining an implicit prior on the side. While the description sticks to binary events, it gets quickly more advanced with the notion of partial and Markov exchangeability. With the most interesting connection between those exchangeabilities and sufficiency. (I would however disagree with the statement that “Bayes was the father of parametric Bayesian analysis” [p.133] as this is extrapolating too much from the Essay.) My next remark may be non-sensical, but I would have welcomed an entry at the end of the chapter on cases where the exchangeability representation fails, for instance those cases when there is no sufficiency structure to exploit in the model. A bonus to the chapter is a description of Birkhoff’s ergodic theorem “as a generalisation of de Finetti” (p..134-136), plus half a dozen pages of appendices on more technical aspects of de Finetti’s theorem.

“We want random sequences to pass all tests of randomness, with tests being computationally implemented”. (p.151)

The eighth chapter (“algorithmic randomness”) comes (again!) as a surprise as it centres on the character of Per Martin-Löf who is little known in statistics circles. (The chapter starts with a picture of him with the iconic Oberwolfach sculpture in the background.) Martin-Löf’s work concentrates on the notion of randomness, in a mathematical rather than probabilistic sense, and on the algorithmic consequences. I like very much the section on random generators. Including a mention of our old friend RANDU, the 16 planes random generator! This chapter connects with Chapter 4 since von Mises also attempted to define a random sequence. To the point it feels slightly repetitive (for instance Jean Ville is mentioned in rather similar terms in both chapters). Martin-Löf’s central notion is computability, which forces us to visit Turing’s machine. And its role in the undecidability of some logical statements. And Church’s recursive functions. (With a link not exploited here to the notion of probabilistic programming, where one language is actually named Church, after Alonzo Church.) Back to Martin-Löf, (I do not see how his test for randomness can be implemented on a real machine as the whole test requires going through the entire sequence: since this notion connects with von Mises’ Kollektivs, I am missing the point!) And then Kolmororov is brought back with his own notion of complexity (which is also Chaitin’s and Solomonov’s). Overall this is a pretty hard chapter both because of the notions it introduces and because I do not feel it is completely conclusive about the notion(s) of randomness. A side remark about casino hustlers and their “exploitation” of weak random generators: I believe Jeff Rosenthal has a similar if maybe simpler story in his book about Canadian lotteries.

“Does quantum mechanics need a different notion of probability? We think not.” (p.180)

The penultimate chapter is about Boltzmann and the notion of “physical chance”. Or statistical physics. A story that involves Zermelo and Poincaré, And Gibbs, Maxwell and the Ehrenfests. The discussion focus on the definition of probability in a thermodynamic setting, opposing time frequencies to space frequencies. Which requires ergodicity and hence Birkhoff [no surprise, this is about ergodicity!] as well as von Neumann. This reaches a point where conjectures in the theory are yet open. What I always (if presumably naïvely) find fascinating in this topic is the fact that ergodicity operates without requiring randomness. Dynamical systems can enjoy ergodic theorem, while being completely deterministic.) This chapter also discusses quantum mechanics, which main tenet requires probability. Which needs to be defined, from a frequency or a subjective perspective. And the Bernoulli shift that brings us back to random generators. The authors briefly mention the Einstein-Podolsky-Rosen paradox, which sounds more metaphysical than mathematical in my opinion, although they get to great details to explain Bell’s conclusion that quantum theory leads to a mathematical impossibility (but they lost me along the way). Except that we “are left with quantum probabilities” (p.183). And the chapter leaves me still uncertain as to why statistical mechanics carries the label statistical. As it does not seem to involve inference at all.

“If you don’t like calling these ignorance priors on the ground that they may be sharply peaked, call them nondogmatic priors or skeptical priors, because these priors are quite in the spirit of ancient skepticism.” (p.199)

And then the last chapter (“induction”) brings us back to Hume and the 18th Century, where somehow “everything” [including statistics] started! Except that Hume’s strong scepticism (or skepticism) makes induction seemingly impossible. (A perspective with which I agree to some extent, if not to Keynes’ extreme version, when considering for instance financial time series as stationary. And a reason why I do not see the criticisms contained in the Black Swan as pertinent because they savage normality while accepting stationarity.) The chapter rediscusses Bayes’ and Laplace’s contributions to inference as well, challenging Hume’s conclusion of the impossibility to finer. Even though the representation of ignorance is not unique (p.199). And the authors call again for de Finetti’s representation theorem as bypassing the issue of whether or not there is such a thing as chance. And escaping inductive scepticism. (The section about Goodman’s grue hypothesis is somewhat distracting, maybe because I have always found it quite artificial and based on a linguistic pun rather than a logical contradiction.) The part about (Richard) Jeffrey is quite new to me but ends up quite abruptly! Similarly about Popper and his exclusion of induction. From this chapter, I appreciated very much the section on skeptical priors and its analysis from a meta-probabilist perspective.

There is no conclusion to the book, but to end up with a chapter on induction seems quite appropriate. (But there is an appendix as a probability tutorial, mentioning Monte Carlo resolutions. Plus notes on all chapters. And a commented bibliography.) Definitely recommended!

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]