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Inferring Probabilities of Probabilities

Assume an event \(E\) with an unknown probability \(P(E)\). Given that the event \(E\) has occurred, find the probability distribution of \(P(E)\).

In other words, there is an event whose probability is unknown. We know that the event happens on the very first trial. Based only on this information, how likely is it that the event always happens with a \(50\%\) probability? How about \(0\%\), or \(100\%\)? Find a probability distribution to describe this.


This problem was presented to me by a friend, who had thought of it but didn't know how to solve it. I tried, but I don't have a solution either, and I am interested to see what kind of techniques can be used to solve it.

Note by Ben Frankel
3 years, 7 months ago

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If we start with an existing prior set of probabilities for P(E), we can use Bayesian learning to update them after every trial. However even in this approach the challenge remains to define the prior probabilities. Anindya Bhattacharjee · 3 years, 7 months ago

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@Anindya Bhattacharjee Well, some things are immediately evident about the distribution of \(P(E)\).

Denote the desired probability distribution by \(g(x) = P(P(E) = x | E)\).

  • \(g(0) = 0\)
  • \(\int_{0}^{1}\: g(x)\:\textrm{d}x = 1\)
  • \(a > b \:\Rightarrow\: g(a) > g(b)\)
Ben Frankel · 3 years, 7 months ago

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