## Bertrand-Borel debate

**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.

May 6, 2019 at 5:01 am

With regard to your second point, sensible choices for the statistic in question can certainly be made when only one parameter is unknown. For example, if a univariate sufficient statistic exists then that is naturally the statistic.

Applying the standard hypothesis testing approach to multiparameter problems of course faces more sizeable obstacles. This is probably the reason why Mayo tends to steer away from discussing this latter issue.

P.S. If (in your 2nd sentence) you are classifying both Bertrand and Borel as being “B” (grade?) mathematicians, then I would be happy being in the Z category (!)

May 6, 2019 at 9:37 am

B: I did not realise it could have this interpretation. Certainly not my intent!

May 7, 2019 at 4:30 am

Maintenant je suis intrigué. C’était une faute de frappe, non?

May 7, 2019 at 7:29 am

No, no, I was playing with the irrelevant fact that Bayes, Bertrand, Borel all had a B as their initial letter.