Colin Blyth (1922-2019)

While reading the IMS Bulletin (of March 2020), I found out that Canadian statistician Colin Blyth had died last summer. While we had never met in person, I remember his very distinctive and elegant handwriting in a few letters he sent me, including the above I have kept (along with an handwritten letter from Lucien Le Cam!). It contains suggestions about revising our Is Pitman nearness a reasonable criterion?, written with Gene Hwang and William Strawderman and which took three years to publish as it was deemed somewhat controversial. It actually appeared in JASA with discussions from Malay Ghosh, John Keating and Pranab K Sen, Shyamal Das Peddada, C. R. Rao, George Casella and Martin T. Wells, and Colin R. Blyth (with a much stronger wording than in the above letter!, like “What can be said but “It isn’t I, it’s you that are crazy?”). While I had used some of his admissibility results, including the admissibility of the Normal sample average in dimension one, e.g. in my book, I had not realised at the time that Blyth was (a) the first student of Erich Lehmann (b) the originator of [the name] Simpson’s paradox, (c) the scribe for Lehmann’s notes that would eventually lead to Testing Statistical Hypotheses and Theory of Point Estimation, later revised with George Casella. And (d) a keen bagpipe player and scholar.

maximum likelihood: an introduction

“Basic Principle 0. Do not trust any principle.” L. Le Cam (1990)

Here is the abstract of a International Statistical Rewiew 1990 paper by Lucien Le Cam on maximum likelihood. ISR keeping a tradition of including an abstract in French for every paper, Le Cam (most presumably) wrote his own translation [or maybe wrote the French version first], which sounds much funnier to me and so I cannot resist posting both, pardon my/his French! [I just find “Ce fait” rather unusual, as I would have rather written “Ceci fait”…]:

Maximum likelihood estimates are reported to be best under all circumstances. Yet there are numerous simple examples where they plainly misbehave. One gives some examples for problems that had not been invented for the purpose of annoying maximum likelihood fans. Another example, imitated from Bahadur, has been specially created with just such a purpose in mind. Next, we present a list of principles leading to the construction of good estimates. The main principle says that one should not believe in principles but study each problem for its own sake.

L’auteur a ouï dire que la méthode du maximum de vraisemblance est la meilleure méthode d’estimation. C’est bien vrai, et pourtant la méthode se casse le nez sur des exemples bien simples qui n’avaient pas été inventés pour le plaisir de montrer que la méthode peut être très désagréable. On en donne quelques-uns, plus un autre, imité de Bahadur et fabriqué exprès pour ennuyer les admirateurs du maximum de vraisemblance. Ce fait, on donne une savante liste de principes de construction de bons estimateurs, le principe principal étant qu’il ne faut pas croire aux principes.

The entire paper is just as witty, as in describing the mixture model as “contaminated and not fit to drink”! Or in “Everybody knows that taking logarithms is unfair”. Or, again, in “biostatisticians, being complicated people, prefer to work out not with the dose y but with its logarithm”… And a last line: “One possibility is that there are too many horse hairs in e”.

beware, nefarious Bayesians threaten to take over frequentism using loss functions as Trojan horses!

“It is not a coincidence that textbooks written by Bayesian statisticians extol the virtue of the decision-theoretic perspective and then proceed to present the Bayesian approach as its natural extension.” (p.19)

“According to some Bayesians (see Robert, 2007), the risk function does represent a legitimate frequentist error because it is derived by taking expectations with respect to [the sampling density]. This argument is misleading for several reasons.” (p.18)

During my R exam, I read the recent arXiv posting by Aris Spanos on why “the decision theoretic perspective misrepresents the frequentist viewpoint”. The paper is entitled “Why the Decision Theoretic Perspective Misrepresents Frequentist Inference: ‘Nuts and Bolts’ vs. Learning from Data” and I found it at the very least puzzling…. The main theme is the one caricatured in the title of this post, namely that the decision-theoretic analysis of frequentist procedures is a trick brought by Bayesians to justify their own procedures. The fundamental argument behind this perspective is that decision theory operates in a “for all θ” referential while frequentist inference (in Spanos’ universe) is only concerned by one θ, the true value of the parameter. (Incidentally, the “nuts and bolt” refers to the only case when a decision-theoretic approach is relevant from a frequentist viewpoint, namely in factory quality control sampling.)

“The notions of a risk function and admissibility are inappropriate for frequentist inference because they do not represent legitimate error probabilities.” (p.3)

“An important dimension of frequentist inference that has not been adequately appreciated in the statistics literature concerns its objectives and underlying reasoning.” (p.10)

“The factual nature of frequentist reasoning in estimation also brings out the impertinence of the notion of admissibility stemming from its reliance on the quantifier ‘for all’.” (p.13)

One strange feature of the paper is that Aris Spanos seems to appropriate for himself the notion of frequentism, rejecting the choices made by (what I would call frequentist) pioneers like Wald, Neyman, “Lehmann and LeCam [sic]”, Stein. Apart from Fisher—and the paper is strongly grounded in neo-Fisherian revivalism—, the only frequentists seemingly finding grace in the eyes of the author are George Box, David Cox, and George Tiao. (The references are mostly to textbooks, incidentally.) Modern authors that clearly qualify as frequentists like Bickel, Donoho, Johnstone, or, to mention the French school, e.g., Birgé, Massart, Picard, Tsybakov, none of whom can be suspected of Bayesian inclinations!, do not appear either as satisfying those narrow tenets of frequentism. Furthermore, the concept of frequentist inference is never clearly defined within the paper. As in the above quote, the notion of “legitimate error probabilities” pops up repeatedly (15 times) within the whole manifesto without being explicitely defined. (The closest to a definition is found on page 17, where the significance level and the p-value are found to be legitimate.) Aris Spanos even rejects what I would call the von Mises basis of frequentism: “contrary to Bayesian claims, those error probabilities have nothing to to do with the temporal or the physical dimension of the long-run metaphor associated with repeated samples” (p.17), namely that a statistical  procedure cannot be evaluated on its long term performance…