AP Statistics Curriculum 2007 Bayesian Other
From Socr
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) = Px (1 − P)n − x , (x = 0, 1, …, n)
p(x|P) is proportional to Px(1 − P)n − x
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 + 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) = (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 =
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) = 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
Gamma ( , β + n)
The posterior mean E[λ] approaches the maximum likelihood estimate in the limit as α and β approach 0.