second and higher-order probabilities

[lolgarithms]

Why do I never hear of "second-order probabilities" (probability that the probability of an event is x. for example, you pick a box at random out of 3 boxes, and each box can have either 1, 3 or 5 red marbles out of six marbles)? can measurement or calculation of probability not be affected by random and inaccurate data?

[EnumaElish]

How is your concept different from joint probability? How is it different from conditional probability?

Alternatively, you can define higher-order probability as follows. Let P = Prob{X < x} = CDF(x). Since X is a random variable, so is P (because it is a function of a random variable, with CDF of X as the link function). As a random variable, P is distributed uniformly over [0,1]. Defined this way, higher-order probabilities are not interesting: they are all distributed uniformly over [0,1].

[mXSCNT]

Decision theory does sometimes use second-order probabilities, although they are distrusted by some.

google (http://www.google.com/search?q=second+order+probabilities&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a) has the answers.