Difference with respect to the discrete case

Figure: Examples of variable changes starting from a uniform distribution (``$K$''): A) $Y=0.5\,X+0.25$; B) $Y=\sqrt X$; C) $Y=X^2$; D) $Y=X^4$.

The title of this section is similar to that of Section , but the problem and the conclusions will be different. There we said that the Indifference Principle (or, in its refined modern version, the Maximum Entropy Principle) was a good choice. Here there are problems with infinities and with the fact that it is possible to map an infinite number of points contained in a finite region onto an infinite number of points contained in a larger or smaller finite region. This changes the probability density function. If, moreover, the transformation from one set of variables to the other is not linear (see, e.g., Fig. ), what is uniform in one variable ($X$) is not uniform in another variable (e.g. $Y=X^2$). This problem does not exist in the case of discrete variables, since if $X=x_i$ has a probability $f(x_i)$ then $Y=x_i^2$ has the same probability. A different way of stating the problem is that the Jacobian of the transformation squeezes or stretches the metrics, changing the probability density function.

We will not enter into the open discussion about the optimal choice of the distribution. Essentially we shall use the uniform distribution, being careful to employ the variable which ``seems'' most appropriate for the problem, but You may disagree -- surely with good reason -- if You have a different kind of experiment in mind.

The same problem is also present, but well hidden, in the maximum likelihood method. For example, it is possible to demonstrate that, in the case of normally distributed likelihoods, a uniform distribution of the mean $\mu$ is implicitly assumed (see Section ). There is nothing wrong with this, but one should be aware of it.