Bertrand-Borel debate

On her blog, Deborah Mayo briefly mentioned the Bertrand-Borel debate on the (in)feasibility of hypothesis testing, as reported [and translated] by Erich Lehmann. A first interesting feature is that both [starting with] B mathematicians discuss the probability of causes in the Bayesian spirit of Laplace. With Bertrand considering that the prior probabilities of the different causes are impossible to set and then moving all the way to dismiss the use of probability theory in this setting, nipping the p-values in the bud..! And Borel being rather vague about the solution probability theory has to provide. As stressed by Lehmann.

“The Pleiades appear closer to each other than one would naturally expect. This statement deserves thinking about; but when one wants to translate the phenomenon into numbers, the necessary ingredients are lacking. In order to make the vague idea of closeness more precise, should we look for the smallest circle that contains the group? the largest of the angular distances? the sum of squares of all the distances? the area of the spherical polygon of which some of the stars are the vertices and which contains the others in its interior? Each of these quantities is smaller for the group of the Pleiades than seems plausible. Which of them should provide the measure of implausibility? If three of the stars form an equilateral triangle, do we have to add this circumstance, which is certainly very unlikely apriori, to those that point to a cause?” Joseph Bertrand (p.166)

 

“But whatever objection one can raise from a logical point of view cannot prevent the preceding question from arising in many situations: the theory of probability cannot refuse to examine it and to give an answer; the precision of the response will naturally be limited by the lack of precision in the question; but to refuse to answer under the pretext that the answer cannot be absolutely precise, is to place oneself on purely abstract grounds and to misunderstand the essential nature of the application of mathematics.” Emile Borel (Chapter 4)

Another highly interesting objection of Bertrand is somewhat linked with his conditioning paradox, namely that the density of the observed unlikely event depends on the choice of the statistic that is used to calibrate the unlikeliness, which makes complete sense in that the information contained in each of these statistics and the resulting probability or likelihood differ to an arbitrary extend, that there are few cases (monotone likelihood ratio) where the choice can be made, and that Bayes factors share the same drawback if they do not condition upon the entire sample. In which case there is no selection of “circonstances remarquables”. Or of uniformly most powerful tests.

severe testing : beyond Statistics wars?!

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!

reading pile for X break

severe testing or severe sabotage? [not a book review]

Last week, I received this new book of Deborah Mayo, which I was looking forward reading and annotating!, but thrice alas, the book had been sabotaged: except for the preface and acknowledgements, the entire book is printed upside down [a minor issue since the entire book is concerned] and with some part of the text cut on each side [a few letters each time but enough to make reading a chore!]. I am thus waiting for a tested copy of the book to start reading it in earnest!

 

“an outstanding paper that covers the Jeffreys-Lindley paradox”…

“This is, in this revised version, an outstanding paper that covers the Jeffreys-Lindley paradox (JLP) in exceptional depth and that unravels the philosophical differences between different schools of inference with the help of the JLP. From the analysis of this paradox, the author convincingly elaborates the principles of Bayesian and severity-based inferences, and engages in a thorough review of the latter’s account of the JLP in Spanos (2013).” Anonymous

I have now received a second round of reviews of my paper, “On the Jeffreys-Lindleys paradox” (submitted to Philosophy of Science) and the reports are quite positive (or even extremely positive as in the above quote!). The requests for changes are directed to clarify points, improve the background coverage, and simplify my heavy style (e.g., cutting Proustian sentences). These requests were easily addressed (hopefully to the satisfaction of the reviewers) and, thanks to the week in Warwick, I have already sent the paper back to the journal, with high hopes for acceptance. The new version has also been arXived. I must add that some parts of the reviews sounded much better than my original prose and I was almost tempted to include them in the final version. Take for instance

“As a result, the reader obtains not only a better insight into what is at stake in the JLP, going beyond the results of Spanos (2013) and Sprenger (2013), but also a much better understanding of the epistemic function and mechanics of statistical tests. This is a major achievement given the philosophical controversies that have haunted the topic for decades. Recent insights from Bayesian statistics are integrated into the article and make sure that it is mathematically up to date, but the technical and foundational aspects of the paper are well-balanced.” Anonymous

Deborah Mayo’s talk in Montréal (JSM 2013)

As posted on her blog, Deborah Mayo is giving a lecture at JSM 2013 in Montréal about why Birnbaum’s derivation of the Strong Likelihood Principle (SLP) is wrong. Or, more accurately, why “WCP entails SLP”. It would have been a great opportunity to hear Deborah presenting her case and I am sorry I am missing this opportunity. (Although not sorry to be in the beautiful Dolomites at that time.) Here are the slides:

Deborah’s argument is the same as previously: there is no reason for the inference in the mixed (or Birnbaumized) experiment to be equal to the inference in the conditional experiment. As previously, I do not get it: the weak conditionality principle (WCP) implies that inference from the mixture output, once we know which component is used (hence rejecting the “and we don’t know which” on slide 8), should only be dependent on that component. I also fail to understand why either WCP or the Birnbaum experiment refers to a mixture (sl.13) in that the index of the experiment is assumed to be known, contrary to mixtures. Thus (still referring at slide 13), the presentation of Birnbaum’s experiment is erroneous. It is indeed impossible to force the outcome of y* if tail and of x* if head but it is possible to choose the experiment index at random, 1 versus 2, and then, if y* is observed, to report (E₁,x*) as a sufficient statistic. (Incidentally, there is a typo on slide 15, it should be “likewise for x*”.)

the anti-Bayesian moment and its passing commented

Here is a comment on our rejoinder “the anti-Bayesian moment and its passing” with Andrew Gelman from Deborah Mayo, comment that could not make it through as a comment:

You assume that I am interested in long-term average properties of procedures, even though I have so often argued that they are at most necessary (as consequences of good procedures), but scarcely sufficient for a severity assessment. The error statistical account I have developed is a statistical philosophy. It is not one to be found in Neyman and Pearson, jointly or separately, except in occasional glimpses here and there (unfortunately). It is certainly not about well-defined accept-reject rules. If N-P had only been clearer, and Fisher better behaved, we would not have had decades of wrangling. However, I have argued, the error statistical philosophy explicates, and directs the interpretation of, frequentist sampling theory methods in scientific, as opposed to behavioural, contexts. It is not a complete philosophy…but I think Gelmanian Bayesians could find in it a source of “standard setting”.

You say “the prior is both a probabilistic object, standard from this perspective, and a subjective construct, translating qualitative personal assessments into a probability distribution. The extension of this dual nature to the so-called “conventional” priors (a very good semantic finding!) is to set a reference … against which to test the impact of one’s prior choices and the variability of the resulting inference. …they simply set a standard against which to gauge our answers.”

I think there are standards for even an approximate meaning of “standard-setting” in science, and I still do not see how an object whose meaning and rationale may fluctuate wildly, even in a given example, can serve as a standard or reference. For what?

Perhaps the idea is that one can gauge how different priors change the posteriors, because, after all, the likelihood is well-defined. That is why the prior and not the likelihood is the camel. But it isn’t obvious why I should want the camel. (camel/gnat references in the paper and response).