Bertrand’s paradox

While progressing through Jaynes’ Probability Theory for my classes next week, I reached the end of Chapter 12 where he proposed a “resolution” of Bertrand’s paradox. Jaynes gives the following description of the paradox:

Bertrand’s problem was stated originally in terms of drawing a straight line `at random’ intersecting a circle (…) we do no violence to the problem if we suppose we are tossing straws onto the circle (…) What is the probability that the chord thus defined [by a random straw] is longer than the side of the inscribed equilateral triangle?

My understanding of the paradox is that it provides a perfect illustration of the lack of meaning of “random” and of the need for a proper definition of the σ-algebra leading to a probabilised space. Different σ-algebras lead to different probabilities, e.g., 1/4, 1/3, 1/2… However, Jaynes considers there is a “correct” answer and endeavours to construct an invariant distribution on the location of the centre of the chord, achieving Borel’s distribution

p(x) = \frac{x}{\sqrt{1-x^2}}

on the chord proportion L/2R. When illustrating this approach, I tried to bring an empirical vision and generated “random” straws by picking both ends at random on the (-10,10)^2 square. Here is a subsample of 10³ such straws interesting with the unit circle. The empirical distribution of the chord proportion is actually quite in agreement with Borel’s distribution, with a probability of being longer than the side of 1/2, as shown below but this does not validate Jaynes’ argument, simply illustrates that I picked the same σ-algebra as his. (Every σ-algebra considered by Bertrand could as well have been used for the simulation.)

5 Responses to “Bertrand’s paradox”

  1. […] for the candidate’s formula). This started me thinking about how this was open to a Bertrand’s paradox of its own. Indeed, randomness for the needle throw can be represented in many […]

  2. […] for the incomprehensible shots at formalised mathematics (Bourbakism), measure theory (as in Bertrand’s paradox) and Feller (although the later’s anti-Bayes stance  may have induced these attacks), I […]

  3. […] may have had reservations about the “randomness” of the straws I plotted to illustrate Bertrand’s paradox. As they were all going North-West/South-East. I had […]

  4. X:

    When I read this (and similar examples) in Jaynes’s book, I interpreted his philosophy somewhat differently. I agree that if you read Jaynes literally, it appears that he claims there is a correct distribution, similarly to the way he appears to claim elsewhere that maximum entropy gives the correct unique specification of a probability distribution given some specified moments.

    But I give a slightly different interpretation. To me, what Jaynes is saying is that given what appears to be partial information, it makes sense to make some strong assumptions based on some theoretical framework, and then go with those assumptions until they’re violated. Once you have some data it should be easy enough to find problems with any particular model you assume. Then when there’s a problem, Jaynes says to go back and see what you did wrong earlier and to specify what additional information you have. To me, this is very consistent with the 3 steps of BDA (model building, inference, model checking); in fact my reading of Jaynes was a major inspiration behind our framing Bayesian data analysis in that way. Even though I expect that Jaynes himself would’ve been horrified by my pragmatic-style Bayesian philosophy, I think that it is in the spirit of the best of his work.

    If you can wait a couple years, perhaps we can do that seminar together during my next année sabbatique!

    • For me the sections on Bertand’s paradox are very clear in that he claims only Emile Borel saw the light in proposing a definitive solution (which happens to coincide with Jaynes’) and all other (from Bertrand to Poincaré, to von Mises) were mathematical nit-pickers… The very final sentence of the section where he mentions an experimental verification of Borel’s law is quite supportive of this interpretation.

      And you are more than welcome to run this seminar next time you visit!!!

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