AP Statistics Curriculum 2007 Bayesian Other

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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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