Bayesian inference for hidden distribution

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This problem can come up in many practical situations where some form of destructive testing is required.



Given: A coin that has probability f of heads and 1-f of tails.



Each time the coin is picked up, f is drawn from an unknown distribution, p(f).



You are allowed to flip the coin n times each time it is picked up, so that it has that same f for n flips, then you must put it down - this is one trial.



When you pick it up again, it has a new f, drawn from p.



The goal is to estimate p. Assume that the prior for p is constant.



For a given value of n, how should the estimate for p(f) be updated after each trial?




probability




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  • Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
    – Theoretical Economist
    7 hours ago










  • This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
    – joriki
    2 hours ago











  • okay - I've edited it to be more clear.
    – IMM
    1 hour ago














up vote
0
down vote

favorite
1












This problem can come up in many practical situations where some form of destructive testing is required.



Given: A coin that has probability f of heads and 1-f of tails.



Each time the coin is picked up, f is drawn from an unknown distribution, p(f).



You are allowed to flip the coin n times each time it is picked up, so that it has that same f for n flips, then you must put it down - this is one trial.



When you pick it up again, it has a new f, drawn from p.



The goal is to estimate p. Assume that the prior for p is constant.



For a given value of n, how should the estimate for p(f) be updated after each trial?




probability




share|cite|improve this question





















  • Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
    – Theoretical Economist
    7 hours ago










  • This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
    – joriki
    2 hours ago











  • okay - I've edited it to be more clear.
    – IMM
    1 hour ago












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





This problem can come up in many practical situations where some form of destructive testing is required.



Given: A coin that has probability f of heads and 1-f of tails.



Each time the coin is picked up, f is drawn from an unknown distribution, p(f).



You are allowed to flip the coin n times each time it is picked up, so that it has that same f for n flips, then you must put it down - this is one trial.



When you pick it up again, it has a new f, drawn from p.



The goal is to estimate p. Assume that the prior for p is constant.



For a given value of n, how should the estimate for p(f) be updated after each trial?




probability




share|cite|improve this question













This problem can come up in many practical situations where some form of destructive testing is required.



Given: A coin that has probability f of heads and 1-f of tails.



Each time the coin is picked up, f is drawn from an unknown distribution, p(f).



You are allowed to flip the coin n times each time it is picked up, so that it has that same f for n flips, then you must put it down - this is one trial.



When you pick it up again, it has a new f, drawn from p.



The goal is to estimate p. Assume that the prior for p is constant.



For a given value of n, how should the estimate for p(f) be updated after each trial?





probability





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 1 hour ago
























asked 7 hours ago









IMM

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  • Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
    – Theoretical Economist
    7 hours ago










  • This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
    – joriki
    2 hours ago











  • okay - I've edited it to be more clear.
    – IMM
    1 hour ago
















  • Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
    – Theoretical Economist
    7 hours ago










  • This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
    – joriki
    2 hours ago











  • okay - I've edited it to be more clear.
    – IMM
    1 hour ago















Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
– Theoretical Economist
7 hours ago




Welcome to Math.SE. Your question needs more context. Please read the guide on how to ask questions and edit your question accordingly.
– Theoretical Economist
7 hours ago












This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
– joriki
2 hours ago





This makes no sense. You say $f$ is drawn from a certain distribution and then ask how that distribution should be updated. The answer is: If that's indeed the distribution from which $f$ is drawn, then it shouldn't be updated, it just is. If, on the other hand, $p(f)$ is some kind of estimate you have for the distribution from which $f$ is drawn, then you need a prior over all possible distributions in order to update it, but you don't mention anything like that. You do mention a "prior for $p$", but then provide not a prior for $p$ but one specific $p$ (which could be a prior for $f$).
– joriki
2 hours ago













okay - I've edited it to be more clear.
– IMM
1 hour ago




okay - I've edited it to be more clear.
– IMM
1 hour ago















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