Second and higher-order probabilities

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SUMMARY

This discussion centers on the concept of second-order probabilities, specifically the probability that the probability of an event equals a certain value. An example provided involves selecting a box from three, each containing a different number of red marbles. The conversation also distinguishes second-order probabilities from joint and conditional probabilities, emphasizing that higher-order probabilities, when defined as a function of a random variable, are uniformly distributed over the interval [0,1]. The use of second-order probabilities in decision theory is acknowledged, though their reliability is questioned.

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  • Understanding of basic probability concepts, including random variables.
  • Familiarity with cumulative distribution functions (CDF).
  • Knowledge of joint and conditional probabilities.
  • Basic principles of decision theory.
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  • Explore the implications of second-order probabilities in decision theory.
  • Study the differences between joint and conditional probabilities in depth.
  • Investigate the properties and applications of cumulative distribution functions (CDF).
  • Learn about the reliability and criticisms of higher-order probabilities in statistical analysis.
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Statisticians, data scientists, and decision theorists interested in advanced probability concepts and their applications in real-world scenarios.

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

google has the answers.
 

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