The competent frequentist would presumably not be befuddled by these supposed paradoxes. Since he would not be befuddled (or so I am fairly certain), the “paradoxes” fail to prove the superiority of the Bayesian approach. Frankly, the treatment of these “paradoxes” in terms of repeated experiments seems to straightforward that I don’t know how you can possibly think there’s a problem.
Say you have a circle. On this circle you draw the inscribed equilateral triangle.
Simple, right?
Okay. For a random chord in this circle, what is the probability that the chord is longer than the side in the triangle?
So, to choose a random chord, there are three obvious methods:
Pick a point on the circle perimeter, and draw the triangle with that point as an edge. Now when you pick a second point on the circle perimeter as the other endpoint of your chord, you can plainly see that in 1⁄3 of the cases, the resulting chord will be longer than the triangles’ side.
Pick a random radius (line from center to perimeter). Rotate the triangle so one of the sides bisect this radius. Now you pick a point on the radius to be the midpoint of your chord. Apparently now, the probability of the chord being longer than the side is 1⁄2.
Pick a random point inside the circle to be the midpoint of your chord (chords are unique by midpoint). If the midpoint of a chord falls inside the circle inscribed by the triangle, it is longer than the side of the triangle. The inscribed circle has an area 1⁄4 of the circumscribing circle, and that is our probability.
WHAT NOW?!
The solution is to choose the distribution of chords that lets us be maximally indifferent/ignorant. I.e. the one that is both scale, translation and rotation invariant (i.e. invariant under Affine transformations). The second solution has those properties.
The competent frequentist would presumably not be befuddled by these supposed paradoxes. Since he would not be befuddled (or so I am fairly certain), the “paradoxes” fail to prove the superiority of the Bayesian approach. Frankly, the treatment of these “paradoxes” in terms of repeated experiments seems to straightforward that I don’t know how you can possibly think there’s a problem.
Say you have a circle. On this circle you draw the inscribed equilateral triangle.
Simple, right?
Okay. For a random chord in this circle, what is the probability that the chord is longer than the side in the triangle?
So, to choose a random chord, there are three obvious methods:
Pick a point on the circle perimeter, and draw the triangle with that point as an edge. Now when you pick a second point on the circle perimeter as the other endpoint of your chord, you can plainly see that in 1⁄3 of the cases, the resulting chord will be longer than the triangles’ side.
Pick a random radius (line from center to perimeter). Rotate the triangle so one of the sides bisect this radius. Now you pick a point on the radius to be the midpoint of your chord. Apparently now, the probability of the chord being longer than the side is 1⁄2.
Pick a random point inside the circle to be the midpoint of your chord (chords are unique by midpoint). If the midpoint of a chord falls inside the circle inscribed by the triangle, it is longer than the side of the triangle. The inscribed circle has an area 1⁄4 of the circumscribing circle, and that is our probability.
WHAT NOW?!
The solution is to choose the distribution of chords that lets us be maximally indifferent/ignorant. I.e. the one that is both scale, translation and rotation invariant (i.e. invariant under Affine transformations). The second solution has those properties.
Wikipedia article)