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Question about Bayesian Inference, Posterior Distribution
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I have a posterior probability of $p_i$ which is based on a Beta prior and some data from a binomial distribution:
I have another procedure:
$$P(E)=prod_i in I p_i^k_i(1-p_i)^1-k_i$$
which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find $P(E)$?
UPDATE: I think the issue may be the notation. $P(E)$ should actually be $P(E|p_1,dotsc,p_i,dotsc,p_)$. Then if we are looking for the marginal probability, $P(E)$, then we need to solve for
$$int dotsi int_0^1 P(E|p_1,dotsc,p_i,dotsc,p_)P(p_1,,dotsc,p_i,dotsc,p_), dp_i.$$
Because the $p_i$'s are all independent, we can probably simplify the question a lot.
probability-theory bayesian
edited Jul 22 at 20:41
Daniel Buck
2,3041623
asked Jul 22 at 0:43
bob
114
add a comment |Â
up vote
1
down vote
favorite
I have a posterior probability of $p_i$ which is based on a Beta prior and some data from a binomial distribution:
I have another procedure:
$$P(E)=prod_i in I p_i^k_i(1-p_i)^1-k_i$$
which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find $P(E)$?
UPDATE: I think the issue may be the notation. $P(E)$ should actually be $P(E|p_1,dotsc,p_i,dotsc,p_)$. Then if we are looking for the marginal probability, $P(E)$, then we need to solve for
$$int dotsi int_0^1 P(E|p_1,dotsc,p_i,dotsc,p_)P(p_1,,dotsc,p_i,dotsc,p_), dp_i.$$
Because the $p_i$'s are all independent, we can probably simplify the question a lot.
probability-theory bayesian
edited Jul 22 at 20:41
Daniel Buck
2,3041623
asked Jul 22 at 0:43
bob
114
Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have a posterior probability of $p_i$ which is based on a Beta prior and some data from a binomial distribution:
I have another procedure:
$$P(E)=prod_i in I p_i^k_i(1-p_i)^1-k_i$$
which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find $P(E)$?
UPDATE: I think the issue may be the notation. $P(E)$ should actually be $P(E|p_1,dotsc,p_i,dotsc,p_)$. Then if we are looking for the marginal probability, $P(E)$, then we need to solve for
$$int dotsi int_0^1 P(E|p_1,dotsc,p_i,dotsc,p_)P(p_1,,dotsc,p_i,dotsc,p_), dp_i.$$
Because the $p_i$'s are all independent, we can probably simplify the question a lot.
probability-theory bayesian
edited Jul 22 at 20:41
Daniel Buck
2,3041623
asked Jul 22 at 0:43
bob
114
I have a posterior probability of $p_i$ which is based on a Beta prior and some data from a binomial distribution:
I have another procedure:
$$P(E)=prod_i in I p_i^k_i(1-p_i)^1-k_i$$
which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find $P(E)$?
UPDATE: I think the issue may be the notation. $P(E)$ should actually be $P(E|p_1,dotsc,p_i,dotsc,p_)$. Then if we are looking for the marginal probability, $P(E)$, then we need to solve for
$$int dotsi int_0^1 P(E|p_1,dotsc,p_i,dotsc,p_)P(p_1,,dotsc,p_i,dotsc,p_), dp_i.$$
Because the $p_i$'s are all independent, we can probably simplify the question a lot.
probability-theory bayesian
edited Jul 22 at 20:41
Daniel Buck
2,3041623
asked Jul 22 at 0:43
bob
114
edited Jul 22 at 20:41
Daniel Buck
2,3041623
edited Jul 22 at 20:41
Daniel Buck
2,3041623
edited Jul 22 at 20:41
Daniel Buck
2,3041623
2,3041623
asked Jul 22 at 0:43
bob
114
asked Jul 22 at 0:43
bob
114
asked Jul 22 at 0:43
bob
114
114
Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32
add a comment |Â
Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32
Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32
Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32
add a comment |Â
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Is the idea that you're interested in the joint distribution of some subset of the Bernoulli random variables within the Binomial distribution? Not really sure what you're trying to ask here.
– Kevin Li
Jul 24 at 13:32