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Consider Jaynes' solution to the Bertrand paradox using the principle of indifference. Why doesn't a similar argument apply to the Borel-Kolmogorov paradox?

Is there something wrong with arguing that since the problem does not specify an orientation for the sphere, rotating the sphere should not affect the resulting distribution arrived at by the chosen limiting process?

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    $\begingroup$ Given that this is a non-mathematical argument, you can always use it! And equally always find someone arguing against it...! $\endgroup$
    – Xi'an
    Commented Jul 26, 2012 at 7:45
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    $\begingroup$ Also I do not thing Jaynes' argument closes the debate about Bertrand paradox: there are an infinite number of ways of physically drawing lines at random, as discussed in this post of mine. $\endgroup$
    – Xi'an
    Commented Jul 26, 2012 at 7:50
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    $\begingroup$ Did you notice how that Wikipedia article actually quotes Jaynes on the B-K paradox? "… the term 'great circle' is ambiguous until we specify what limiting operation is to produce it. The intuitive symmetry argument presupposes the equatorial limit; yet one eating slices of an orange might presuppose the other." It seems to me this answers your question. $\endgroup$
    – whuber ♦
    Commented Jul 26, 2012 at 12:41
  • $\begingroup$ @whuber: I took that to mean that the question-asker had to specify the limiting process. I didn't think it meant that the principle of indifference could be used to force a unique choice in the limiting process. Is that how you see the statement? $\endgroup$
    – Neil G
    Commented Jul 26, 2012 at 16:54
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    $\begingroup$ @whuber: Lol :) Okay, well I'm still trying to understand it. Jaynes writes that the maximum entropy principle and Jeffreys' priors are extensions of the principle of indifference, and those are pretty convincing to me. So, there seems to be something interesting here. $\endgroup$
    – Neil G
    Commented Jul 26, 2012 at 20:56

2 Answers 2

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One one hand, we have a pre-theoretic, intuitive understanding of probability. On the other, we have Kolomogorov's formal axiomatization of probability.

The principle of indifference belongs to our intuitive understanding of probability. We feel that any formalization of probability should respect it. However, as you note, our formal theory of probability does not always do this, and the the Borel-Komogorov paradox is one of the cases where it doesn't.

So, here's what I think you're really asking: How do we resolve the conflict between this attractive intuitive principle and our modern measure-theoretic theory of probability?

One could side with our formal theory, as the other answer and the commenters do. They claim that, if you choose the limit to the equator in the Borel-Kolmogorov paradox in a certain way, the principle of indifference does not hold, and our intuitions are incorrect.

I find this unsatisfactory. I believe that if our formal theory does not capture this basic and obviously true intuition, then it is deficient. We should seek to modify the theory, not reject this basic principle.

Alan Hájek, a philosopher of probability, has taken this position, and he argues convincingly for it in this article. A longer article by him on conditional probability can be found here, where he also discusses some classic problems like the two envelopes paradox.

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I don't see the point of the "principle of indifference". The Wikipedia article's answer is better: "Probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined." In other words, without even restricting ourselves to questions of probability, "An ambiguously-posed question does not have a single unambiguous answer."

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  • $\begingroup$ Thanks for your answer. Did you read Jaynes' defence of the principle of indifference? E. Jaynes, “Where do we stand on Maximum Entropy?,” R. Levine and M. Tribus, Eds. The MIT Press, 1979, pp. 15–118. $\endgroup$
    – Neil G
    Commented Aug 6, 2012 at 16:00

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