Archive for Jerzy Neyman

Seminal ideas and controversies in Statistics [book review]

Posted in Books, Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on May 24, 2025 by xi'an

CRC Press sent CHANCE this book for review. Since the topic was of clear interest to me, with an author who significantly contributed to the field—my only recollection meeting Roderick Little was during the Australian Statistical Conference in Adelaïde, in 2012, at the start of my Oz 2012 Tour!—, I took the opportunity of the nearest weekend to browse through Seminal ideas and controversies in Statistics. I like very much the idea of selecting a dozen key papers in the history of Statistics and of discussing why. In fact, this reminded me of my classics seminar, which lasted the few years I was 100% in charge of the Master program in Dauphine (and which I hope I could restart!). Checking the list of the papers I then suggested my students, I see some overlap with 9 papers out of the 15 groups. (I also remember Steve Fienberg making suggestions for that list, while he was spending a sabbatical in Paris at CREST.) Given that community of focus and purpose, and contrary to my wont, I have really very little of substance to criticize or wish about the book. The less when reading the following

“On a personal note, I met Yates [author of a 1984 paper on tests for 2×2 contingency tables discussing the relevance of conditioning on one or both margins], a charming man, when I was a young graduate student who knew next to nothing about statistics; we discussed the joys of traversing the Cuillin Ridge in Skye.”

since completing that ridge remains high in my mountain-climbing bucket-list! (Possibly next year, since we are running an ICMS workshop on the Island.)

The first paper in the series is more than a foundational paper since (The) Fisher’s 1922 paper is about creating (almost) ex nihilo the field of (modern) mathematical statistics. I don’t know if there is any equivalence in other scientific disciplines of such an impact (and of such a man)… Roderick Little manages to convincingly engage with Fisher’s dismissive views on (not yet called) Bayesian analysis, although, to the latter’s defence, the formalisation of Bayesian inference at that time had not yet emerged. The second chapter is discussing Yates’ 1984 paper on tests for 2×2 contingency tables that he wrote 50 years after writing the original one in the first volume of JRSS. Roderick Little adds a detailed Bayesian analysis with the three standard reference priors, Jeffreys’ version proving quite close to Fisher’s exact test (conditional on both margins). The third chapter is aiming at the generic challenge of hypothesis testing, from the well-known opposition between Fisher and Neyman (both on the cover), to questioning the sanity of hard-set thresholds (with a mention of our American Statistician call to abandon (shi)p!). The later (thus) refers to the recent literature on the replicability crisis and the now famous ASA statement on p-values by Ron Wasserstein and Nicole Lazar, analysed in the chapter. But I would have like to read another full section on alternatives to hypothesis testing. While now a niche interest (imho), Fisher’s attempt at creating a posterior distribution without a prior, aka fiducial inference, is discussed in Chapter 4 with the Behrens-Fisher problem as the illustrating example. The chapter feels rather anticlimactic, with the comparison relying on the (Malay) Ghosh and Kim (2001) simulation results.

Birnbaum’s (1962) likelihood principle is the topic of Chapter 5 (and I cannot remember any of my students choosing this paper over the years, although there was at least one). Roderick Little recalls some sentences from the JASA discussion as an appetiser, a reminder of the time when these discussions could turn in scathing attacks. The chapter contains excerpts from Berger and Wolpert (1988)—which they were writing while I was spending a year at Purdue and which I have always recommended to my PhD students, albeit not for the classic seminar. It then moves to the controversies that surround this principle since its inception, in particular those accumulated by Deborah Mayo (also on the cover) as reported on the ‘Og. In the recent years, I have become less excited about the LP, in part due to the imprecision in its statement, which opens the door to conflicting interpretations. And in part due to the scarcity of models with non-trivial sufficient statistics. (I am also wondering if the sufficiency issue we highlighted in our ABC model choice criticism does relate to the mixture example at the end of the chapter.)

The next chapter is one all for compromise, through the calibrated Bayes perspective that credible statements should be close to confidence statements in the long run. Which I remember him presenting at ASC 2012. The concept is found in the very 1984 paper by Don Rubin (also on the cover) that contains the concept behind Approximate Bayesian Computation (ABC). And the chapter proceeds by listing strengths and weaknesses of frequentist and Bayesian perspectives, towards a fusion of both., e.g. though posterior predictive checks.

While the choice of a (general public) paper from Scientific American may sound surprising in Chapter 7, with Efron’s (on the cover) and Morris’ 1977 Stein’s paradox, I cannot but applaud, the more because this was the first paper I read when starting my PhD on the James-Stein estimators. Although this may sound like happening eons ago, the James and Stein (1961) paper—which is my age!—”created a considerable backlash” by toppling unbiasedness from its pedestal and exhibiting a paradox that 1+1+1≠3… Which Little reinterprets via a random effect (or Bayesian hierarchical) model. (And a chapter where I learned that Little’s father was a journalist, a characteristic he shared with Bruce Lindsay, as I found at Blonde, Glasgow, during an ICMS workshop). Relatedly, the next chapter is about the “57 varieties [of regression] paper” by Demptster, Schatzoff and Wermuth (1977). Apparently connected with Heinz 57 varieties of pickles. The paper considers Stein and ridge and variable selections versions for variable selection. The chapter also covers (Bayesian) Lasso and BART, as well as a brief all too brief mention of Spike & Slab priors—with my friend Veronika Ročková missing from the authors’ index!—,  but I was expecting from the title other, robust, forms of regression like L¹ regression and econometrics digressions. Chapter 10 can however been seen as a proxy since covering generalized estimating equations from a 1986 Biometrika paper of Liang and Zeger, with no Bayesian aspect (and an expected appearance of Communications in Statistics B).

Chapter 9 covers the almost immediately classic 1995 paper of Benjamini and Hochbeg on multiple regressions (that Series B turned into a discussion paper ten years later!). Although it spends more time on Berry’s (2012) recommendations than on FDR. The computational Chapter 11 brings together Efron’s (1979) bootstrap [with his picture on the cover] and MCMC, represented by the founding paper of Gelfand and Smith (1990, if mistakenly set in 1988 on p140). A bit of a strange mix imho as the former is more inferential than computational. And not giving the EM algorithm that much space. And not questioning MCMC methods as a good proxy to posterior distributions. Tukey’s Future of Data Analysis (as founding exploratory data analysis) and Breiman’s Two cultures (as launching statistical machine learning) meet in Chapter 12. (With a reminder that the latter invokes Occam’s razor—which may not be that appropriate for hugely overparameterised machine learning black boxes—and…the Rashomon principle! Meaning that distinct models may all fit the same data. Let me nitpickingly add the reference to Ryûnosuke Akutagawa as the author of Rashômon and other stories that Kurosawa adapted in his splendid movie). The chapter contains critical remarks from David Cox, Brad Efron, David Bickel, and Andrew Gelman, with a further section on Little’s view on modelling.

The last three chapters are on design and sampling, in connection with Little’s (and Rubin’s) works in the area. With a 1934 paper of Neyman (whose picture on the cover could have been chosen differently, albeit no fault of Neyman [or of Little!] that his toothbrush style of moustache dramatically got out of fashion!). With a return to calibrated Bayes and a reminiscence of Little’s time at the World Fertility Survey but (apparently) no mention of the probabilistic aspects of modern censuses (that saw my friends Steve Fienberg on the one side and Larry Brown and Marty Wells on the other side argue for and against it!), again relating to the reliance on statistical models. Chapter 14 relates randomized clinical trials to causality, which makes a (worthy) appearance there. Roderick Little also makes a clear case there against the retracted study linking vaccines and autism, a call that will unlikely not reach the current Trump administration and its Secretary of Health.

The book concludes with a list of twenty style and grammar suggestions for improved writing.

As should be crystal-clear from the above, I quite enjoyed the book and would definitely use its reading list in a graduate course whenever the opportunity arises. Once again, some choices are more personal to the author than others, and I would have place more emphasis on the fantastic Dawid, Stone and Zidek (1973)—with Jim Zidek also missing from the author index—, but all make sense in a walk through statistical classics. Let me however regret the absence therein of major actors like, e.g., D. Blackwell, C.R. Rao,  or G. Wahba (except in a stylistic example p199), two of whom were awarded the International Prize in Statistics.

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

Fisher, Bayes, and predictive Bayesian inference [seminar]

Posted in Statistics with tags , , , , , , , , , on April 4, 2021 by xi'an

An interesting Foundations of Probability seminar at Rutgers University this Monday, at 4:30ET, 8:30GMT, by Sandy Zabell (the password is Angelina’s birthdate):

R. A. Fisher is usually perceived to have been a staunch critic of the Bayesian approach to statistics, yet his last book (Statistical Methods and Scientific Inference, 1956) is much closer in spirit to the Bayesian approach than the frequentist theories of Neyman and Pearson.  This mismatch between perception and reality is best understood as an evolution in Fisher’s views over the course of his life.  In my talk I will discuss Fisher’s initial and harsh criticism of “inverse probability”, his subsequent advocacy of fiducial inference starting in 1930, and his admiration for Bayes expressed in his 1956 book.  Several of the examples Fisher discusses there are best understood when viewed against the backdrop of earlier controversies and antagonisms.

severe testing : beyond Statistics wars?!

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on January 7, 2019 by xi'an

A timely start to my reading Deborah Mayo’s [properly printed] Statistical Inference as Severe Testing (How to get beyond the Statistics Wars) on the Armistice Day, as it seems to call for just this, an armistice! And the opportunity of a long flight to Oaxaca in addition… However, this was only the start and it took me several further weeks to peruse seriously enough the book (SIST) before writing the (light) comments below. (Receiving a free copy from CUP and then a second one directly from Deborah after I mentioned the severe sabotage!)

Indeed, I sort of expected a different content when taking the subtitle How to get beyond the Statistics Wars at face value. But on the opposite the book is actually very severely attacking anything not in the line of the Cox-Mayo severe testing line. Mostly Bayesian approach(es) to the issue! For instance, Jim Berger’s construct of his reconciliation between Fisher, Neyman, and Jeffreys is surgically deconstructed over five pages and exposed as a Bayesian ploy. Similarly, the warnings from Dennis Lindley and other Bayesians that the p-value attached with the Higgs boson experiment are not probabilities that the particle does not exist are met with ridicule. (Another go at Jim’s Objective Bayes credentials is found in the squared myth of objectivity chapter. Maybe more strongly than against staunch subjectivists like Jay Kadane. And yet another go when criticising the Berger and Sellke 1987 lower bound results. Which even extends to Vale Johnson’s UMP-type Bayesian tests.)

“Inference should provide posterior probabilities, final degrees of support, belief, probability (…) not provided by Bayes factors.” (p.443)

Another subtitle of the book could have been testing in Flatland given the limited scope of the models considered with one or at best two parameters and almost always a Normal setting. I have no idea whatsoever how the severity principle would apply in more complex models, with e.g. numerous nuisance parameters. By sticking to the simplest possible models, the book can carry on with the optimality concepts of the early days, like sufficiency (p.147) and and monotonicity and uniformly most powerful procedures, which only make sense in a tiny universe.

“The estimate is really a hypothesis about the value of the parameter.  The same data warrant the hypothesis constructed!” (p.92)

There is an entire section on the lack of difference between confidence intervals and the dual acceptance regions, although the lack of unicity in defining either of them should come as a bother. Especially outside Flatland. Actually the following section, from p.193 onward, reminds me of fiducial arguments, the more because Schweder and Hjort are cited there. (With a curve like Fig. 3.3. operating like a cdf on the parameter μ but no dominating measure!)

“The Fisher-Neyman dispute is pathological: there’s no disinterring the truth of the matter (…) Fisher grew to renounce performance goals he himself had held when it was found that fiducial solutions disagreed with them.”(p.390)

Similarly the chapter on the “myth of the “the myth of objectivity””(p.221) is mostly and predictably targeting Bayesian arguments. The dismissal of Frank Lad’s arguments for subjectivity ends up [or down] with a rather cheap that it “may actually reflect their inability to do the math” (p.228). [CoI: I once enjoyed a fantastic dinner cooked by Frank in Christchurch!] And the dismissal of loss function requirements in Ziliak and McCloskey is similarly terse, if reminding me of Aris Spanos’ own arguments against decision theory. (And the arguments about the Jeffreys-Lindley paradox as well.)

“It’s not clear how much of the current Bayesian revolution is obviously Bayesian.” (p.405)

The section (Tour IV) on model uncertainty (or against “all models are wrong”) is somewhat limited in that it is unclear what constitutes an adequate (if wrong) model. And calling for the CLT cavalry as backup (p.299) is not particularly convincing.

It is not that everything is controversial in SIST (!) and I found agreement in many (isolated) statements. Especially in the early chapters. Another interesting point made in the book is to question whether or not the likelihood principle at all makes sense within a testing setting. When two models (rather than a point null hypothesis) are X-examined, it is a rare occurrence that the likelihood factorises any further than the invariance by permutation of iid observations. Which reminded me of our earlier warning on the dangers of running ABC for model choice based on (model specific) sufficient statistics. Plus a nice sprinkling of historical anecdotes, esp. about Neyman’s life, from Poland, to Britain, to California, with some time in Paris to attend Borel’s and Lebesgue’s lectures. Which is used as a background for a play involving Bertrand, Borel, Neyman and (Egon) Pearson. Under the title “Les Miserables Citations” [pardon my French but it should be Les Misérables if Hugo is involved! Or maybe les gilets jaunes…] I also enjoyed the sections on reuniting Neyman-Pearson with Fisher, while appreciating that Deborah Mayo wants to stay away from the “minefields” of fiducial inference. With, mot interestingly, Neyman himself trying in 1956 to convince Fisher of the fallacy of the duality between frequentist and fiducial statements (p.390). Wisely quoting Nancy Reid at BFF4 stating the unclear state of affair on confidence distributions. And the final pages reawakened an impression I had at an earlier stage of the book, namely that the ABC interpretation on Bayesian inference in Rubin (1984) could come closer to Deborah Mayo’s quest for comparative inference (p.441) than she thinks, in that producing parameters producing pseudo-observations agreeing with the actual observations is an “ability to test accordance with a single model or hypothesis”.

“Although most Bayesians these days disavow classic subjective Bayesian foundations, even the most hard-nosed. “we’re not squishy” Bayesian retain the view that a prior distribution is an important if not the best way to bring in background information.” (p.413)

A special mention to Einstein’s cafe (p.156), which reminded me of this picture of Einstein’s relative Cafe I took while staying in Melbourne in 2016… (Not to be confused with the Markov bar in the same city.) And a fairly minor concern that I find myself quoted in the sections priors: a gallimaufry (!) and… Bad faith Bayesianism (!!), with the above qualification. Although I later reappear as a pragmatic Bayesian (p.428), although a priori as a counter-example!

X-Outline of a Theory of Statistical Estimation

Posted in Books, Statistics, University life with tags , , , , , , , , , , on March 23, 2017 by xi'an

While visiting Warwick last week, Jean-Michel Marin pointed out and forwarded me this remarkable paper of Jerzy Neyman, published in 1937, and presented to the Royal Society by Harold Jeffreys.

“Leaving apart on one side the practical difficulty of achieving randomness and the meaning of this word when applied to actual experiments…”

“It may be useful to point out that although we are frequently witnessing controversies in which authors try to defend one or another system of the theory of probability as the only legitimate, I am of the opinion that several such theories may be and actually are legitimate, in spite of their occasionally contradicting one another. Each of these theories is based on some system of postulates, and so long as the postulates forming one particular system do not contradict each other and are sufficient to construct a theory, this is as legitimate as any other. “

This paper is fairly long in part because Neyman starts by setting Kolmogorov’s axioms of probability. This is of historical interest but also needed for Neyman to oppose his notion of probability to Jeffreys’ (which is the same from a formal perspective, I believe!). He actually spends a fair chunk on explaining why constants cannot have anything but trivial probability measures. Getting ready to state that an a priori distribution has no meaning (p.343) and that in the rare cases it does it is mostly unknown. While reading the paper, I thought that the distinction was more in terms of frequentist or conditional properties of the estimators, Neyman’s arguments paving the way to his definition of a confidence interval. Assuming repeatability of the experiment under the same conditions and therefore same parameter value (p.344).

“The advantage of the unbiassed [sic] estimates and the justification of their use lies in the fact that in cases frequently met the probability of their differing very much from the estimated parameters is small.”

“…the maximum likelihood estimates appear to be what could be called the best “almost unbiassed [sic]” estimates.”

It is also quite interesting to read that the principle for insisting on unbiasedness is one of producing small errors, because this is not that often the case, as shown by the complete class theorems of Wald (ten years later). And that maximum likelihood is somewhat relegated to a secondary rank, almost unbiased being understood as consistent. A most amusing part of the paper is when Neyman inverts the credible set into a confidence set, that is, turning what is random in a constant and vice-versa. With a justification that the credible interval has zero or one coverage, while the confidence interval has a long-run validity of returning the correct rate of success. What is equally amusing is that the boundaries of a credible interval turn into functions of the sample, hence could be evaluated on a frequentist basis, as done later by Dennis Lindley and others like Welch and Peers, but that Neyman fails to see this and turn the bounds into hard values. For a given sample.

“This, however, is not always the case, and in general there are two or more systems of confidence intervals possible corresponding to the same confidence coefficient α, such that for certain sample points, E’, the intervals in one system are shorter than those in the other, while for some other sample points, E”, the reverse is true.”

The resulting construction of a confidence interval is then awfully convoluted when compared with the derivation of an HPD region, going through regions of acceptance that are the dual of a confidence interval (in the sampling space), while apparently [from my hasty read] missing a rule to order them. And rejecting the notion of a confidence interval being possibly empty, which, while being of practical interest, clashes with its frequentist backup.

reading classics (#9)

Posted in Books, Statistics, University life with tags , , , , , , , , on February 24, 2013 by xi'an

In today’s classics seminar, my student Bassoum Abou presented the 1981 paper written by Charles Stein for the Annals of Statistics, Estimating the mean of a normal distribution, recapitulating the advances he made on Stein estimators, minimaxity and his unbiased estimator of risk. Unfortunately; this student missed a lot about paper and did not introduce the necessary background…So I am unsure at how much the class got from this great paper… Here are his slides (watch out for typos!)

 Historically, this paper is important as this is one of the very few papers published by Charles Stein in a major statistics journal, the other publications being made in conference proceedings. It contains the derivation of the unbiased estimator of the loss, along with comparisons with posterior expected loss.