# AP Statistics Curriculum 2007 Bayesian Other

### From Socr

(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> | ||

+ | |||

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) | ||

+ | |||

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 05:45, 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) = *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*^{α − 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 *P*^{α + x − 1}(1 − *P*)^{β + n − x − 1}

The posterior distribution of the Binomial is

(P|x) ~ β(α + x, β + n – x)