Editing Principle of indifference (section)

==Application to continuous variables==
Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables. A typical case of misuse is the following example:
*Suppose there is a cube hidden in a box. A label on the box says the cube has a side length between 3 and 5&nbsp;cm.
*We don't know the actual side length, but we might assume that all values are equally likely and simply pick the mid-value of 4&nbsp;cm.
*The information on the label allows us to calculate that the surface area of the cube is between 54 and 150&nbsp;cm<sup>2</sup>. We don't know the actual surface area, but we might assume that all values are equally likely and simply pick the mid-value of 102&nbsp;cm<sup>2</sup>.
*The information on the label allows us to calculate that the volume of the cube is between 27 and 125&nbsp;cm<sup>3</sup>. We don't know the actual volume, but we might assume that all values are equally likely and simply pick the mid-value of 76&nbsp;cm<sup>3</sup>.
*However, we have now reached the impossible conclusion that the cube has a side length of 4&nbsp;cm, a surface area of 102&nbsp;cm<sup>2</sup>, and a volume of 76&nbsp;cm<sup>3</sup>!

In this example, mutually contradictory estimates of the length, surface area, and volume of the cube arise because we have assumed three mutually contradictory distributions for these parameters: a [[uniform distribution (continuous)|uniform distribution]] for any one of the variables implies a non-uniform distribution for the other two. In general, the principle of indifference does not indicate which variable (e.g. in this case, length, surface area, or volume) is to have a uniform epistemic [[probability distribution]].

Another classic example of this kind of misuse is the [[Bertrand paradox (probability)|Bertrand paradox]]. [[Edwin T. Jaynes]] introduced the [[principle of transformation groups]], which can yield an epistemic probability distribution for this problem. This generalises the principle of indifference, by saying that one is indifferent between ''equivalent problems'' rather than indifferent between propositions. This still reduces to the ordinary principle of indifference when one considers a permutation of the labels as generating equivalent problems (i.e. using the permutation transformation group). To apply this to the above box example, we have three random variables related by geometric equations. If we have no reason to favour one trio of values over another, then our prior probabilities must be related by the rule for changing variables in continuous distributions. Let ''L'' be the length, and ''V'' be the volume. Then we must have
:<math>f_L(L) = \left|{\partial V \over \partial L}\right|f_V(V)=3 L^{2} f_V(L^{3})</math>,

where <math>f_L,\,f_V</math> are the [[probability density function]]s (pdf) of the stated variables. This equation has a general solution: <math>f(L) = {K \over L}</math>, where ''K'' is a normalization constant, determined by the range of ''L'', in this case equal to:
:<math>K^{-1}=\int_{3}^{5}{dL \over L} = \log\left({5 \over 3}\right)</math>

To put this "to the test", we ask for the probability that the length is less than 4. This has probability of:
:<math>Pr(L<4)=\int_{3}^{4}{dL \over L \log({5 \over 3})}= {\log({4 \over 3}) \over \log({5 \over 3})} \approx 0.56</math>.

For the volume, this should be equal to the probability that the volume is less than 4<sup>3</sup> = 64. The pdf of the volume is
:<math>f(V^{{1 \over 3}}) {1 \over 3} V^{-{2 \over 3}}={1 \over 3 V \log({5 \over 3})}</math>.

And then probability of volume less than 64 is
:<math>Pr(V<64)=\int_{27}^{64}{dV \over 3 V \log({5 \over 3})}={\log({64 \over 27}) \over 3 \log({5 \over 3})}={3 \log({4 \over 3}) \over 3 \log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})} \approx 0.56</math>.

Thus we have achieved invariance with respect to volume and length. One can also show the same invariance with respect to surface area being less than 6(4<sup>2</sup>) = 96. However, note that this probability assignment is not necessarily a "correct" one. For the exact distribution of lengths, volume, or surface area will depend on how the "experiment" is conducted.

The fundamental hypothesis of [[statistical physics]], that any two microstates of a system with the same total energy are equally probable at [[thermodynamic equilibrium|equilibrium]], is in a sense an example of the principle of indifference. However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under ''which'' parameterization the probability density will be uniform. [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] justifies the use of canonically [[conjugate variable]]s, such as positions and their conjugate momenta.

The [[wine/water paradox]] shows a dilemma with linked variables, and which one to choose.