Difference between revisions of "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 01:40, 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) = \({n \choose x}\) \( P^x \) \((1 - P)^{n - x}\) , (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)
Bayesian Inference for the Poisson Distribution
A discrete random variable x is said to have a Poisson distribution of mean \(\lambda\) if it has the density
P(x|\(\lambda\)) = (\(\lambda^x</x!\))\(e^{-\lambda}\)
Supose that you have n observations x=(x1, x2, …, xn) from such a distribution so that the likelihood is
L(\(\lambda\)|x) = \(\lambda^T e^{(-n \lambda)}\), where T = \(\sum{k_i}\)
In Bayesian inference, the conjugate prior for the parameter \(\lambda\) of the Poisson distribution is the Gamma distribution.
\(\lambda \sim\) Gamma(\(\alpha\) , \(\beta\) )
The Poisson parameter \(\lambda\) is distributed accordingly to the parameterized Gamma density g in terms of a shape and inverse scale parameter \(\alpha\) and \(\beta\) respectively
g(\(\lambda\)|\(\alpha\) , \(\beta\)) = \(\displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)}\) \(\lambda^{\alpha - 1} e^{-\beta \lambda}\)
For \(\lambda\) > 0
Then, given the same sample of n measured values \(k_i\) from our likelihood and a prior of Gamma(\(\alpha\), \(\beta\)), the posterior distribution becomes
\(\lambda \sim\) Gamma (\(\alpha + \displaystyle\sum_{i=1}^{\infty} k_i\) , \(\beta\) + n)
The posterior mean E[\(\lambda\)] approaches the maximum likelihood estimate in the limit as \(\alpha\) and \(\beta\) approach 0.
