Archive for hypothesis testing

A modern introduction to probability and statistics [book review]

Posted in Books, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , on July 12, 2025 by xi'an

In the plane to Bengaluru, I read through the book A modern introduction to probability and statistics, by Graham Upton—whose Measuring Animal Abundance I reviewed for CHANCE a while ago—, which is based on the earlier Understanding Statistics, written jointly with Ian Cook. (Not to be confused with A modern introduction to probability and statistics by Dekking et al.) The subtitle is understanding statistical principles in the computer age. Sorry, in the age of the computer. While the cover is most pleasant (and modern), as noticed by an AF flight attendant, the contents are very very standard and could have been written decades ago since the main concession to “the” computer age is the inclusion of a few R commands at the end of most chapters. There are even a few distribution tables here and there (in case “the” computer is not available). But there is no other connection with computational statistics or statistical computing.

The classicism of the contents and the intended audience mean there is little therein on which to either object or criticise. The mixture of elementary probability and basic statistics in a single textbook always feels awkward to me and I think I would have trouble teaching solely from this material. Apart from the glaring typo on the variance of the sum of two correlated random variables on page 87, missing the factor 2 in front of the covariance, while correct(ed) p97 (and the inevitable “the the” typo spotted once). My main criticisms are on the potential confusion between samples and populations in the early chapters, when some statistics are used as motivational examples, as for instance in a (hidden) Monte Carlo stabilisation to the limiting values (p57), way before the Law of Large Numbers is introduced,, the variable mileage in mathematical rigour (while being uncertain that first year students can handle integrals and derivatives), the textbook examples, and the amount of the book contents spent on descriptive statistics and even more on the “classical” tests, with no critical perspective on using point nulls or p-values. The book concludes with a four page (benevolent) chapter on Bayesian statistics that is superfluous imho, or even counterproductive since my experience with a rushed introduction to Bayesian principles almost always result in a rejection of said principles. Plus, the illustration with the coin tossing is not particularly helpful since Andrew maintains that one can load a die, but cannot bias a coin. (A similar reservation on the half-page 289 coverage on pseudo-random generation and Monte Carlo principles for computing p-values.)

Minor (mostly idiosyncratic) remarks follow: CLT prior to LLN,   n-1 in sample sd, little to no model criticism (ntbcf goodness of fit), missing an opportunity when mentioning the varying probability of a day being a birthday (p31) in contrast with BDA cover story, and another opportunity to cite the 2024 Ig Nobel Prize for coin tossing around the LLN, an unclear definition for random variables( p53) and a potentially confusing introduction of Poisson distributions through a informal reference to Poisson processes (and no reason why the years of accession of the kings of Sussex and England till Guillaume—making a return on p178 with the Domesday Book—in 1066 should follow such a process as suggested in Figure 3.5), a surprising definition of the constant e as the special case of exp(x) when x=1 and its series expansion (p70), omitting proofs on laws of sums of iid rv’s by introducing moment generating functions rather late, another obscure reference to a 16th German treatise on surveying as a precursor of the CLT (p131), a proof for the normalising constant of the Normal density that will most likely escape most first year students, a introduction of the t, F, and χ² distributions with no mention of their respective densities (pp141-147), never defining a joint Normal distribution density, insisting on unbiasedness without noting that maximum likelihood—with a strange motivation that it “makes the next sample of n observations most likely to resemble the data in the current sample (p228)—estimators are almost always biased, an abundance of footnotes that may prove of little interest for the youngest readers.

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

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.]

Nature (statistical) tidbits

Posted in Statistics with tags , , , , , , , , , , , , , , , , , , , , , , , , , , on January 3, 2025 by xi'an

In the 28 November issue of Nature, with this black wallaby left with little to survive after the massive wildfires of 2020, which I read on my way to Nice, several entries related with statistics at large:

Nice meeting!

Posted in pictures, R, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , , on December 18, 2024 by xi'an

The ICSDS 2024 meeting in Nice is quite impressive and not primarily because it is in Nice under a beautiful December sun. As other (numerous) IMS meetings I attended (since the initial one in Uppsala in 1990!), the program is of high quality and along topics that are currently moving fast or emerging. From the sessions I attended, e-values are strongly represented, although it remains unclear to me why they should constitute a major departure from p-values, as they stick to hypothesis testing, Type I error, power, and the whole paraphernalia of Neyman-Pearson formalism. If I manage to attend a BIRS workshop on the subject next Summer, I may manage to get a better e-derstanding!The MCMC (only!) session included a presentation by Guanyang Wang that generalised different approximate MCMC schemes into a unified one. And one by Filippo Ascolani on Gibbs beating the competition! I also attended the Bayesian prediction session, where my friends Sonia Petrone and Chris Holmes have presentations on their respective Series B papers. I discussed both on the ‘Og, on 15 March 2023 and 07 November 2022, respectively. This time, I found that both talks had a Bayesian bootstrap flavour, which is not surprising when considering the non-parametric nature of the approach. And they left me wondering at it being protected from overfitting.
My only plenary session was Cynthia Dwork’s on outcome indistinguishability, which, while related to the privacy topics I was topic, remained somewhat obscure as to its purpose. Meaning I have to get through the paper to get a more holistic perspective.
Of course, Nice in Winter is a very nice place, with the waterfront available for running an uninterrupted 15km as we found out with Jérémie Houssineau (at a brisk 4’09” pace I had not planned before starting!) and the sea all for myself (for a dozen minutes before losing digits!). Unfortunately I had to skip the final day due to examinations of the Paris Dauphine MASH master. And miss Stan receiving a student award. But I am looking forward the next iterations of ICSDS. (Not including Copenhagen, Madrid and many many other places in 2025, since ICSDS seemed a most common name for conferences, some presumably predatory! The true location is Sevilla, to keep up with the Mediterranean theme of ICSDS!)

mixture models [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , on August 14, 2024 by xi'an

Strangely enough, I became aware of this new book on mixtures through one of these annoying emails “Your work has been cited n times this week“… Mixture Models (Parametric, Semiparametric, and New Directions) by Weixin Yao and Sijia Wang got published by CRC Press earlier this year, within the Monographs on Statistics and Applied Probability green series (#175), and covers across 380 pages most aspects of mixture (and hidden Markov) estimation, if with strong emphasis on maximum likelihood estimation, while the new directions are unsurprisingly those pursued by the authors, namely robust and semi-parametric estimation, as well as model selection by testing.

An early warning about this book review is that I co-edited a Handbook of Mixture Analysis with my friends Sylvia Früwirth-Schnatter and Gilles Celeux a few years ago. I am therefore biased in what I would have included in a new book on the topic, the more because I find the available literature already plentiful, even though the early (1984) book of Titterington et al. that was my entry to the field may have become an historical reference. For instance, Finite Mixtures by McLachlan and Peel (2000) remains relevant, with similar emphasis on maximum likelihood and the EM algorithm, while Sylvia’s Finite Mixture and Markov Switching Models is still a reference to this day.

And an additional warning on me not being a massive fan of semi- and non-parametric estimation in this setting…

Preliminaries that may explain my limited enthusiasm about the book and its limited originality. Not that I found significant errors there (even though “improper priors [do not always] yield improper posteriors” [p.145] as we demonstrated in several papers), however, I had trouble with the uneven pace adopted by the authors that often skim some topics of importance while spending an inconsiderate amount of space on less relevant once. Some items get many bibliographical references, while others do not. For instance, EM receives a lion’s share (see, e..g, Sections 6.6 and 6.7). Or the 12 pages of proof in Chapter 10. Declination of sections into mixtures, mixtures of regressions, multivariate mixtures, hidden Markov models, and so on feels somewhat repetitive. This is particularly the case for the “mixture regression models” chapter.

The book also contains Bayesian entries, with a first introduction (p.105) in the discrete data chapter that precedes the short Bayesian chapter #4 (p.145), the same issue arising for related algorithms like Gibbs (p.107) that “estimate properties of the joint posterior” and MCMC (p.112). Which sort of erases the specificity of a Bayesian approach by reducing it to one item in the toolbox (with the wrong stress on MAP estimates). In this Bayesian chapter, MCMC validation is handled for discrete state spaces while applied in general spaces. The focus is mostly on relabelling for the following label switching chapter, albeit a large collection of methods are compared if not mentioned.

Handing an unknown number of components by hypothesis testing is supported in the next short chapter, although very little is said about reversible jump MCMC. And there is no general discussion on the consistency of these tests, in particular with bootstrap. Or at least on the regularity conditions they request. An puzzling paradox (p.191) is the existence of an unbounded Fisher information of an exponential mixture

\pi\mathcal Exp(1)+(1-\pi)\mathcal Exp(2)

when the weight π is the parameter (and close to 1).

High-dimensional mixtures in Chapter 8 are mostly handled by linear projections in smaller subspaces, which is natural given that they preserve the mixture structure but open a Pandora box of a wide range of proposed methods, again with little comparison available. Except in the R final section opposing several R functions on the same dataset (if unconclusively).

The semi-parametric chapters mention Dirichlet process priors, albeit briefly, but fail to relate to the recent works on using these when inferring about the number of components. Or failing to do so. There is also a very limited connection pointed out with machine learning but little can be gathered from the three page presentation (pp.308-310). These chapters also have significant overlap with the review paper of Xiang et al. (2019) in Statistical Science.

Most chapters end up with an R section, which usually reads as a quick demo of a related R package, like BayesLCA or our own mixtool. Hence not massively helpful beyond pointers to these packages. The numerical illustrations also are unevenly distributed between chapters, from nothing at all to four pages of small font tables on an MSE comparison between more or less robust approaches undertaken by Yu et al. (2020).

The above thus explains why I am not particularly excited about this bibliographical addition to the analysis of mixtures. It does offer a reference for researchers in the field by adding recent references and approaches to the existing books mentioned above, but I could not recommend it as a textbook (as suggested on p.xiii).

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