AP Statistics Curriculum 2007 Bayesian Other

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)