Archive for John Maynard Keynes

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!]

abstract for “Bayes’ Theorem: then and now”

Posted in Books, Mountains, Statistics, Travel, University life with tags , , , , , , , , , on March 19, 2013 by xi'an

Here is my abstract for the invited talk I will give at EMS 2013 in Budapest this summer (the first two banners were sites of EMS 2013 conferences as well, which came above the European Meeting of Statisticians on a Google search for EMS 2013):

What is now called Bayes’ Theorem was published and maybe mostly written by Richard Price in 1763, 250 ago. It was re-discovered independently (?) in 1773 by Pierre Laplace, who put it to good use for solving statistical problems, launching what was then called inverse probability and now goes under the name of Bayesian statistics. The talk will cover some historical developments of Bayesian statistics, focussing on the controversies and disputes that marked and stil mark its evolution over those 250 years, up to now. It will in particular address some arguments about prior distributions made by John Maynard Keynes and Harold Jeffreys, as well as divergences about the nature of testing by Dennis Lindley, James Berger, and current science philosophers like Deborah Mayo and Aris Spanos, and misunderstandings on Bayesian computational issues, including those about approximate Bayesian computations (ABC).

I was kindly asked by the scientific committee of EMS 2013 to give a talk on Bayes’ theorem: then and now, which suited me very well for several reasons: first, I was quite interested in giving an historical overview, capitalising on earlier papers about Jeffreys‘ and Keynes‘ books, my current re-analysis of the Jeffreys-Lindley’s paradox, and exchanges around the nature of Bayesian inference. (As you may guess from the contents of the abstract, even borrowing from the article about Price in Significance!) Second, the quality of the programme is definitely justifying attending the whole conference. And not only for meeting again with many friends. At last, I have never visited Hungary and this is a perfect opportunity for starting my summer break there!

someone who might benefit from increased contacts with the statistical community

Posted in Books, Statistics with tags , , , , , on July 23, 2012 by xi'an

A (kind of automated) email I got today:

Your name has come to our attention as someone who might benefit from increased contacts with the international statistical community. Given your professional interests and your statistical background (noting your publication ‘Reading Keynes’ Treatise on Probability’ in the journal International Statistical Review, volume 79, 2011), you should consider elected membership in the International Statistical Institute (ISI).

Hmmm, thanks but no thanks, I am not certain I need become a member of the ISI to increase my contacts with the international statistical community! (Disclaimer: This post makes fun of the anonymous emailing, not of the ISI!)

epidemiology in Le Monde

Posted in Books, Statistics, University life with tags , , , , , , , , , on February 19, 2012 by xi'an

Quite an interesting weekend Le Monde issue: a fourth (2 pages!) of the science folder is devoted to epidemiology… In the statistical sense. (The subtitle is actually Strengths and limitations of Statistics.) The paper does not delve into technical statistical issues but points out the logical divergence between a case-by-case study and an epidemiological study. The impression that the higher the conditioning (i.e. the more covariates), the better the explanation is a statistical fallacy some of the opponents interviewed in the paper do not grasp. (Which reminded me of Keynes seemingly going the same way.) The short paragraph written on causality and Hill’s criteria is vague enough to concur to the overall remark that causality can never been proved or disproved… The fourth examples illustrating the strengths and limitations are tobacco vs. lung cancer, a clear case except for R.A. Fisher!, mobile phones vs. brain tumors, a not yet conclusive setting, hepatitis B vaccine vs. sclerosis, lacking data (the pre-2006 records were destroyed for legal reasons), and leukemia vs. nuclear plants, with a significant [?!] correlation between the number of cases and the distance to a nuclear plant. (The paper was inspired by a report recently published by the French Académie de Médecine on epidemiology in France.) The science folder also includes a review of a recent Science paper by Wilhite and Fong on the coercive strategies used by some journals/editors to increase their impact factor, e.g., “you cite Leukemia [once in 42 references]. Consequently, we kindly ask you to add references of articles published in Leukemia to your present article”.

Error and Inference [#4]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , on September 21, 2011 by xi'an

(This is the fourth post on Error and Inference, again and again yet being a raw and naïve reaction following a linear and slow reading of the book, rather than a deeper and more informed criticism.)

‘The defining feature of an inductive inference is that the premises (evidence statements) can be true while the conclusion inferred may be false without a logical contradiction: the conclusion is “evidence transcending”.”—D. Mayo and D. Cox, p.249, Error and Inference, 2010

The seventh chapter of Error and Inference, entitled “New perspectives on (some old) problems of frequentist statistics“, is divided in four parts, written by David Cox, Deborah Mayo and Aris Spanos, in different orders and groups of authors. This is certainly the most statistical of all chapters, not a surprise when considering that David Cox is involved, and I thus have difficulties to explain why it took me so long to read through it…. Overall, this chapter is quite important by its contribution to the debate on the nature of statistical testing.

‘The advantage in the modern statistical framework is that the probabilities arise from defining a probability model to represent the phenomenon of interest. Had Popper made use of the statistical testing ideas being developed at around the same time, he might have been able to substantiate his account of falsification.”—D. Mayo and D. Cox, p.251, Error and Inference, 2010

The first part of the chapter is Mayo’s and Cox’ “Frequentist statistics as a theory of inductive inference“. It was first published in the 2006 Erich Lehmann symposium. And available on line as an arXiv paper. There is absolutely no attempt there to link of clash with the Bayesian approach, this paper is only looking at frequentist statistical theory as the basis for inductive inference. The debate therein about deducing that H is correct from a dataset successfully facing a statistical test is classical (in both senses) but I [unsurprisingly] remain unconvinced by the arguments. The null hypothesis remains the calibrating distribution throughout the chapter, with very little (or at least not enough) consideration of what happens when the null hypothesis does not hold.  Section 3.6 about confidence intervals being another facet of testing hypotheses is representative of this perspective. The p-value is defended as the central tool for conducting hypothesis assessment. (In this version of the paper, some p’s are written in roman characters and others in italics, which is a wee confusing until one realises that this is a mere typo!)  The fundamental imbalance problem, namely that, in contiguous hypotheses, a test cannot be expected both to most often reject the null when it is [very moderately] false and to most often accept the null when it is right is not discussed there. The argument about substantive nulls in Section 3.5 considers a stylised case of well-separated scientific theories, however the real world of models is more similar to a greyish  (and more Popperian?) continuum of possibles. In connection with this, I would have thought more likely that the book would address on philosophical grounds Box’s aphorism that “all models are wrong”. Indeed, one (philosophical?) difficulty with the p-values and the frequentist evidence principle (FEV) is that they rely on the strong belief that one given model can be exact or true (while criticising the subjectivity of the prior modelling in the Bayesian approach). Even in the typology of types of null hypotheses drawn by the authors in Section 3, the “possibility of model misspecification” is addressed in terms of the low power of an omnibus test, while agreeing that “an incomplete probability specification” is unavoidable (an argument found at several place in the book that the alternative cannot be completely specified).

‘Sometimes we can find evidence for H0, understood as an assertion that a particular discrepancy, flaw, or error is absent, and we can do this by means of tests that, with high probability, would have reported a discrepancy had one been present.”—D. Mayo and D. Cox, p.255, Error and Inference, 2010

The above quote relates to the Failure and Confirmation section where the authors try to push the argument in favour of frequentist tests one step further, namely that that “moderate p-values” may sometimes be used as confirmation of the null. (I may have misunderstood, the end of the section defending a purely frequentist, as in repeated experiments, interpretation. This reproduces an earlier argument about the nature of probability in Section 1.2, as characterising the “stability of relative frequencies of results of repeated trials”) In fact, this chapter and other recent readings made me think afresh about the nature of probability, a debate that put me off so much in Keynes (1921) and even in Jeffreys (1939). From a mathematical perspective, there is only one “kind” of probability, the one defined via a reference measure and a probability, whether it applies to observations or to parameters. From a philosophical perspective, there is a natural issue about the “truth” or “realism” of the probability quantities and of the probabilistic statements. The book and in particular the chapter consider that a truthful probability statement is the one agreeing with “a hypothetical long-run of repeated sampling, an error probability”, while the statistical inference school of Keynes (1921), Jeffreys (1939), and Carnap (1962) “involves quantifying a degree of support or confirmation in claims or hypotheses”, which makes this (Bayesian) sound as less realistic… Obviously, I have no ambition to solve this long-going debate, however I see no reason in the first approach to be more realistic by being grounded on stable relative frequencies à la von Mises. If nothing else, the notion that a test should be evaluated on its long run performances is very idealistic as the concept relies on an ever-repeating, an infinite sequence of identical trials. Relying on probability measures as self-coherent mathematical measures of uncertainty carries (for me) as much (or as less) reality as the above infinite experiment. Now, the paper is not completely entrenched in this interpretation, when it concludes that “what makes the kind of hypothetical reasoning relevant to the case at hand is not the long-run low error rates associated with using the tool (or test) in this manner; it is rather what those error rates reveal about the data generating source or phenomenon” (p.273).

‘If the data are so extensive that accordance with the null hypothesis implies the absence of an effect of practical importance, and a reasonably high p-value is achieved, then it may be taken as evidence of the absence of an effect of practical importance.”—D. Mayo and D. Cox, p.263, Error and Inference, 2010

The paper mentions several times conclusions to be drawn from a p-value near one, as in the above quote. This is an interpretation that does not sit well with my understanding of p-values being distributed as uniforms under the null: very high  p-values should be as suspicious as very low p-values. (This criticism is not new, of course.) Unless one does not strictly adhere to the null model, which brings back the above issue of the approximativeness of any model… I also found fascinating to read the criticism that “power appertains to a prespecified rejection region, not to the specific data under analysis” as I thought this equally applied to the p-values, turning “the specific data under analysis” into a departure event of a prespecified kind.

(Given the unreasonable length of the above, I fear I will continue my snailpaced reading in yet another post!)

Jaynes’ re-read

Posted in Books, Statistics, University life with tags , , , , , , , on March 21, 2011 by xi'an

On many technical issues we disagree strongly with de Finetti. It appears to us that his way of treating infinite sets has opened up a Pandora’s box of useless and unecessary paradoxes.”  E.T. Jaynes, PT, p.xxi

On Friday, despite the cancellation of the reading seminar on Jaynes’ Probability Theory, I completed my slides on Chapters 4 (Elementary hypothesis testing) to 14 (Simple applications of decision theory), plus of course Chapter 20 (Model comparison). I skipped Chapter 15 (Paradoxes of probability theory), despite its extensive and exciting coverage of the marginalisation paradoxes which saw Jaynes opposing David, Stone, and Zidek (and even the whole Establishment, page 470), as it would have taken me another morning at the very least… (Next year, maybe, if the seminar resumes?!)

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Reading seminar on Jaynes’ Probability Theory

Posted in Statistics with tags , , on March 14, 2011 by xi'an

The reading seminar on Jaynes’ Probability Theory is planned to start on next Monday at CREST. (All welcome, just register with Nadine Guedj at ensae.fr) I have so far covered Chapters 4-8 and I need to speed up if I want to keep my schedule. So far, reading carefully through the book has induced neither a burst of criticisms (as it occurred with Keynes’ Treatise on Probability) since there is no strong departure from traditional Bayesian inference, nor a foundational shift in my understanding of the field. The chapter on testing is quite well-written and starting with testing rather than estimation is a brilliant idea. The only annoying part is the permanent bashing of measure theory, which leads to an informal processing of improper priors. Here are the first slides written up to now: