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>
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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 06:32, 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) = {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)


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) = (λx < / 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) = λTe( − nλ), where T = \sum{k_i}

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

\lambda \sim 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) = \displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)} λα − 1e − βλ For λ > 0


Then, given the same sample of n measured values ki from our likelihood and a prior of Gamma(α, β), the posterior distribution becomes


\lambda \sim Gamma (\alpha + \displaystyle\sum_{i=1}^{\infty} k_i , β + n)

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

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