Jaynes' Solution Using The "maximum Ignorance" Principle
In his 1973 paper The Well-Posed Problem, Edwin Jaynes proposed a solution to Bertrand's paradox, based on the principle of "maximum ignorance"—that we should not use any information that is not given in the statement of the problem. Jaynes pointed out that Bertrand's problem does not specify the position or size of the circle, and argued that therefore any definite and objective solution must be "indifferent" to size and position. In other words: the solution must be both scale invariant and translation invariant.
To illustrate: assume that chords are laid at random onto a circle with a diameter of 2, for example by throwing straws onto it from far away. Now another circle with a smaller diameter (e.g., 1.1) is laid into the larger circle. Then the distribution of the chords on that smaller circle needs to be the same as on the larger circle. If the smaller circle is moved around within the larger circle, the probability must not change either. It can be seen very easily that there would be a change for method 3: the chord distribution on the small red circle looks qualitatively different from the distribution on the large circle:
The same occurs for method 1, though it is harder to see in a graphical representation. Method 2 is the only one that is both scale invariant and translation invariant; method 3 is just scale invariant, method 1 is neither.
However, Jaynes did not just use invariances to accept or reject given methods: this would leave the possibility that there is another not yet described method that would meet his common-sense criteria. Jaynes used the integral equations describing the invariances to directly determine the probability distribution. In this problem, the integral equations indeed have a unique solution, and it is precisely what was called "method 2" above, the random radius method.
Read more about this topic: Bertrand Paradox (probability)
Famous quotes containing the words solution, maximum, ignorance and/or principle:
“Any solution to a problem changes the problem.”
—R.W. (Richard William)
“Only at his maximum does an individual surpass all his derivative elements, and become purely himself. And most people never get there. In his own pure individuality a man surpasses his father and mother, and is utterly unknown to them.”
—D.H. (David Herbert)
“To fear death, my friends, is only to think ourselves wise, without being wise: for it is to think that we know what we do not know. For anything that men can tell, death may be the greatest good that can happen to them: but they fear it as if they knew quite well that it was the greatest of evils. And what is this but that shameful ignorance of thinking that we know what we do not know?”
—Socrates (469–399 B.C.)
“A certain secret jealousy of the British Minister is always lurking in the breast of every American Senator, if he is truly democratic; for democracy, rightly understood, is the government of the people, by the people, for the benefit of Senators, and there is always a danger that the British Minister may not understand this political principle as he should.”
—Henry Brooks Adams (1838–1918)