Indeed, the R code does not produce anything. Sorry about this, I cannot trace the original program.

]]>By “this line of argument” I meant both posts, not just this one. I commented here because it was more recent; maybe I should have commented on the original post instead.

The point about introducing more transforms seems fair; Jaynes makes a similar point somewhere, but I can’t remember if it’s in PT:LOS or his original article entitled “The Well-Specified Problem”. But when I try to think of another invariance transform in the Bertrand problem, I can’t come up with anything concrete. Can you find one that would over-specify the problem?

On your devil’s advocacy, I’m not seeing the similarity with frequency, but if one accepts Jaynes’s guiding principle, then the reason one doesn’t condition on the given circle is because there is no specific given circle. Since the problem doesn’t given a circle, one does indeed need a solution that doesn’t pick out one particular circle as special. (But I think I’m just repeating myself here.)

]]>In the maxent section, the argument is basically that to choose a distribution with lower entropy would be to assume information that is not actually available; to choose one with a higher entropy would violate the constraints of the information that is available. The invariance argument is a variation on this theme. Since the problem statement doesn’t mention the size or location of the circle, a probability distribution that purports to solve the problem ought not to depend on these details. Otherwise, the purported solution is assuming information not available in the problem statement.

For example, we know that if the lines follow a distribution that fails scale invariance, then we could inscribe a new circle with the same center as the original circle but with a slightly different radius, compute the distribution of chord lengths with respect to the new circle, and obtain a different distribution than the one with respect to the original circle. If one proposes such a distribution as a basis for calculating the solution to the Bertrand problem, then one is privileging the scale of the original circle, even though there is no information in the problem statement to justify doing so.

Such a distribution is just as “random” as a scale-invariant distribution, which is why Jaynes doesn’t declare the other solutions invalid. But if you’re less concerned with randomness per se and more concerned with avoiding assuming information not in evidence, I think you have good reason to prefer Jaynes’s solution.

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