AP Statistics Curriculum 2007 Bayesian Other

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(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
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'''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:  
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p(P) proportional to <math>P^{α - 1}</math>(1 – P)β – 1 , (P between 0 and 1)
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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
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p(P|x) is proportional to <math>P^{α + x 1}</math>(1 P)^{β + n x 1}
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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 05:43, 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} Px (1 − P)nx , (x = 0, 1, …, n)

p(x|P) is proportional to Px(1 − P)nx


If the prior density has the form: p(P) proportional to Pα − 1(P − 1)β − 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 Pα + x − 1(1 − P)β + nx − 1

The posterior distribution of the Binomial is

(P|x) ~ β(α + x, β + n – x)

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