Difference between revisions of "AP Statistics Curriculum 2007 Bayesian Other"
(New page: Bayesian Inference for the Binomial Distribution The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or fai...) |
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| − | Bayesian Inference for the Binomial Distribution | + | '''Bayesian Inference for the Binomial Distribution |
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The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P | The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P | ||
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p(x|P) is proportional to <math>P^x (1 - P)^{n - x}</math> | p(x|P) is proportional to <math>P^x (1 - P)^{n - x}</math> | ||
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If the prior density has the form: | If the prior density has the form: | ||
| − | p(P) proportional to <math>P^{ | + | p(P) proportional to <math>P^{\alpha - 1}</math><math> (P-1)^{\beta - 1}</math> , (P between 0 and 1) |
then it follows the beta distribution | then it follows the beta distribution | ||
P ~ β(α,β) | P ~ β(α,β) | ||
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From this we can appropriate the posterior which evidently has the form | From this we can appropriate the posterior which evidently has the form | ||
| − | p(P|x) is proportional to <math>P^{ | + | p(P|x) is proportional to <math>P^{\alpha + x - 1}</math><math>(1-P)^{\beta + n - x - 1}</math> |
The posterior distribution of the Binomial is | The posterior distribution of the Binomial is | ||
(P|x) ~ β(α + x, β + n – x) | (P|x) ~ β(α + x, β + n – x) | ||
Revision as of 00:51, 28 May 2009
Bayesian Inference for the Binomial Distribution
The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P
x ~ B(n,P)
subsequently, we can show that p(x|P) = \({n \choose x}\) \( P^x \) \((1 - P)^{n - x}\) , (x = 0, 1, …, n)
p(x|P) is proportional to \(P^x (1 - P)^{n - x}\)
If the prior density has the form:
p(P) proportional to \(P^{\alpha - 1}\)\( (P-1)^{\beta - 1}\) , (P between 0 and 1)
then it follows the beta distribution P ~ β(α,β)
From this we can appropriate the posterior which evidently has the form
p(P|x) is proportional to \(P^{\alpha + x - 1}\)\((1-P)^{\beta + n - x - 1}\)
The posterior distribution of the Binomial is
(P|x) ~ β(α + x, β + n – x)
