Bertrand benchmark meaning9/5/2023 To illustrate: assume that chords are laid at random onto a circle with a diameter of 2, say by throwing straws onto it from far away and converting them to chords by extension/restriction. In other words: the solution must be both scale and translation invariant. 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 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's solution using the "maximum ignorance" principle The three solutions presented by Bertrand correspond to different selection methods, and in the absence of further information there is no reason to prefer one over another accordingly, the problem as stated has no unique solution. The argument is that if the method of random selection is specified, the problem will have a well-defined solution (determined by the principle of indifference). The problem's classical solution (presented, for example, in Bertrand's own work) hinges on the method by which a chord is chosen "at random". In fact, there exists an infinite family of them. Midpoints of the chords chosen at random using method 3 Midpoints of the chords chosen at random using method 2 Midpoints of the chords chosen at random using method 1 Midpoints/chords chosen at random using the above methods. Scatterplots showing simulated Bertrand distributions, The length of the arc is one third of the circumference of the circle, therefore the probability that a random chord is longer than a side of the inscribed triangle is 1 / 3. Observe that if the other chord endpoint lies on the arc between the endpoints of the triangle side opposite the first point, the chord is longer than a side of the triangle. To calculate the probability in question imagine the triangle rotated so its vertex coincides with one of the chord endpoints. Random chords, selection method 1 red = longer than triangle side, blue = shorter The "random endpoints" method: Choose two random points on the circumference of the circle and draw the chord joining them.What is the probability that the chord is longer than a side of the triangle?īertrand gave three arguments (each using the principle of indifference), all apparently valid, yet yielding different results: Suppose a chord of the circle is chosen at random. The Bertrand paradox is generally presented as follows: Consider an equilateral triangle inscribed in a circle. Joseph Bertrand introduced it in his work Calcul des probabilités (1889), as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite. The Bertrand paradox is a problem within the classical interpretation of probability theory. For other paradoxes by Joseph Bertrand, see Bertrand's paradox (disambiguation).
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |