# AP Statistics Curriculum 2007 Bayesian Other

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(P|x) ~ β(α + x, β + n – x) | (P|x) ~ β(α + x, β + n – x) | ||

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+ | '''Bayesian Inference for the Poisson Distribution''' | ||

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+ | A discrete random variable x is said to have a Poisson distribution of mean <math>\lambda</math> if it has the density | ||

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+ | P(x|<math>\lambda</math>) = (<math>\lambda^x</x!</math>)<math>e^{-\lambda}</math> | ||

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+ | Supose that you have n observations x=(x1, x2, …, xn) from such a distribution so that the likelihood is | ||

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+ | L(<math>\lambda</math>|x) = <math>\lambda^T e^{(-n \lambda)}</math>, where T = <math>\sum{k_i}</math> | ||

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+ | In Bayesian inference, the conjugate prior for the parameter <math>\lambda</math> of the Poisson distribution is the Gamma distribution. | ||

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+ | <math>\lambda \sim</math> Gamma(<math>\alpha</math> , <math>\beta</math> ) | ||

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+ | The Poisson parameter <math>\lambda</math> is distributed accordingly to the parameterized Gamma density g in terms of a shape and inverse scale parameter <math>\alpha</math> and <math>\beta</math> respectively | ||

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+ | g(<math>\lambda</math>|<math>\alpha</math> , <math>\beta</math>) = <math>\displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)}</math> <math>\lambda^{\alpha - 1} e^{-\beta \lambda}</math> | ||

+ | For <math>\lambda</math> > 0 | ||

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+ | Then, given the same sample of n measured values <math>k_i</math> from our likelihood and a prior of Gamma(<math>\alpha</math>, <math>\beta</math>), the posterior distribution becomes | ||

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+ | <math>\lambda \sim</math> Gamma (<math>\alpha + \displaystyle\sum_{i=1}^{\infty} k_i</math> , <math>\beta</math> + n) | ||

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+ | The posterior mean E[<math>\lambda</math>] approaches the maximum likelihood estimate in the limit as <math>\alpha</math> and <math>\beta</math> approach 0. |

## Revision as of 06:34, 2 June 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)

**Bayesian Inference for the Poisson Distribution**

A discrete random variable x is said to have a Poisson distribution of mean λ if it has the density

P(x|λ) = (λ^{x} < / *x*!)*e*^{ − λ}

Supose that you have n observations x=(x1, x2, …, xn) from such a distribution so that the likelihood is

L(λ|x) = λ^{T}*e*^{( − nλ)}, where T =

In Bayesian inference, the conjugate prior for the parameter λ of the Poisson distribution is the Gamma distribution.

Gamma(α , β )

The Poisson parameter λ is distributed accordingly to the parameterized Gamma density g in terms of a shape and inverse scale parameter α and β respectively

g(λ|α , β) = λ^{α − 1}*e*^{ − βλ}
For λ > 0

Then, given the same sample of n measured values *k*_{i} from our likelihood and a prior of Gamma(α, β), the posterior distribution becomes

Gamma ( , β + n)

The posterior mean E[λ] approaches the maximum likelihood estimate in the limit as α and β approach 0.