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 Failed to parse (lexing error): P^{α - 1} (1 – P)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 Failed to parse (lexing error): P^{α + x – 1} (1 – P)^{β + n – x – 1}

The posterior distribution of the Binomial is

(P|x) ~ β(α + x, β + n – x)