**Problem:**- Given a circle of radius and a chord drawn randomly on it, what is the probability that the length of the chord is smaller than ?
**Solution 1:**- Choose ``randomly'' two points on the circumference and draw a chord between them: .
**Solution 2:**- Choose a straight line passing through the centre of the circle; then draw a second line, orthogonal to the first, and which intersects it inside the circle at a ``random'' distance from the center: .
**Solution 3:**- Choose ``randomly'' a point inside the circle and draw a straight line orthogonal to the radius that passes through the chosen point .
**Your solution:**- ?
**Question:**- What is the origin of the paradox?
**Answer:**- The problem does not specify how to ``randomly''
choose the chord. The three solutions take a
__uniform__distribution: along the circumference; along the the radius; inside the circle. What is uniform in one variable is not uniform in the others! **Question:**- Which is the
__right__solution?

In fact this kind of paradox, together with abuse of the Indifference
Principle for problems like ``what is the probability that the
sun will rise tomorrow morning'' threw a shadow over
Bayesian methods at the end of the last century. The maximum likelihood
method, which does not make explicit use of prior distributions,
was then seen as a valid solution to the problem. But
in reality
the ambiguity of the proper metrics on which
the initial distribution is uniform has an equivalent
in the arbitrariness of the variable used in the likelihood function.
In the end, what was criticized
when it was stated explicitly in the Bayes formula __is__
accepted passively when it is hidden in the maximum
likelihood method.