AP Statistics Curriculum 2007 Bayesian Other

(Difference between revisions)
 Revision as of 06:04, 26 May 2009 (view source)JayZzz (Talk | contribs) (New page: 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 fai...)← Older edit Revision as of 05:45, 28 May 2009 (view source)JayZzz (Talk | contribs) Newer edit → Line 1: Line 1: - Bayesian Inference for the Binomial Distribution + '''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 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 Line 9: Line 10: p(x|P) is proportional to $P^x (1 - P)^{n - x}$ p(x|P) is proportional to $P^x (1 - P)^{n - x}$ + If the prior density has the form: If the prior density has the form: - p(P) proportional to $P^{α - 1}$(1 – P)β – 1 , (P between 0 and 1) + p(P) proportional to $P^{\alpha - 1} (P-1)^{\beta - 1}$ , (P between 0 and 1) then it follows the beta distribution then it follows the beta distribution P ~ β(α,β) P ~ β(α,β) + From this we can appropriate the posterior which evidently has the form From this we can appropriate the posterior which evidently has the form - p(P|x) is proportional to $P^{α + x – 1}$(1 – P)^{β + n – x – 1} + p(P|x) is proportional to $P^{\alpha + x - 1}(1-P)^{\beta + n - x - 1}$ The posterior distribution of the Binomial is The posterior distribution of the Binomial is (P|x) ~ β(α + x, β + n – x) (P|x) ~ β(α + x, β + n – x)

Revision as of 05:45, 28 May 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) = ${n \choose x}$ 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)