I came across this while working on a persona project. Though the actual problem has to do with estimating article quality on a reddit-like site I think it's best way to explain it like this:
I have a biased coin that comes up heads X% of the time. However I don't know the value X. I know that if I flip it myself I can estimate the probability of any given bias X by using posterior probability density function.
However, let's say that I can't look at the coin when it lands. I have to have an untrustworthy friend check for me. I know my friend will lie to me Y% of the time. Now obviously if he lies exactly 50% of the time I'll never be able to determine anything about the coin. But if Y !=50% then I can.
Given a prior probably density for the coin Pr(X) and a probability density for the trustworthiness of my friend Pr(Y) how can I determine the new probability density for the coin after my friend has reported that he sees it land heads?
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