**O**n 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 tomisunderstand 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.

## Valen in Le Monde

Posted in Books, Statistics, University life with tags blogging, comments, False positive, Le Monde, Monsanto, p-values, Passeur de Sciences, statistical significance, UMPB test, uniformly most powerful tests, Valen Johnson on November 21, 2013 by xi'anValen Johnson made the headline inLe Monde, last week. (More precisely, to the scientific blogPasseur de Sciences. Thanks, Julien, for the pointer!) With the alarming title of “Une étude ébranle un pan de la méthode scientifique”(A study questions one major tool of the scientific approach). The reason for this French fame is Valen’s recent paper in PNAS,Revised standards for statistical evidence, where he puts forward his uniformly most powerful Bayesian tests (recently discussed on the ‘Og) to argue against the standard 0.05 significance level and in favour of “the 0.005 or 0.001 level of significance.”While I do plan to discuss the PNAS paper later (and possibly write a comment letter to PNAS with Andrew), I find interesting the way it made the headlines within days of its (early edition) publication: the argument suggesting to replace .05 with .001 to increase the proportion of reproducible studies is both simple and convincing for a scientific journalist. If only the issue with p-values and statistical testing could be that simple… For instance, the above quote from Valen is reproduced as “an [alternative] hypothesis that stands right below the significance level has in truth only 3 to 5 chances to 1 to be true”, the “truth” popping out of nowhere. (If you read French, the 300+ comments on the blog are also worth their weight in jellybeans…)## Share:

8 Comments »