AP Statistics Curriculum 2007 Bayesian Other
Contents
[hide]Probability and Statistics Ebook - Bayesian Inference for the Binomial and Poisson Distributions
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:
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) \sim P^{\alpha - 1} (P-1)^{\beta - 1}, (P between 0 and 1),
then it follows the beta distribution P \sim \beta(\alpha,\beta).
From this we can appropriate the posterior which evidently has the form: p(P|x) \sim P^{\alpha + x - 1} (1-P)^{\beta + n - x - 1}.
The posterior distribution of the Binomial is (P|x) \sim \beta(\alpha+x,\beta+n-x).
Bayesian Inference for the Poisson Distribution
A discrete random variable x is said to have a Poisson distribution of mean \lambda if it has the density: P(x|\lambda) = {\lambda^x e^{-\lambda}\over x!}
Suppose that you have n observations x=(x_1, x_2, \cdots, x_n) from such a distribution so that the likelihood is: L(\lambda|x) = \lambda^T e^{(-n \lambda)}, where T = \sum_{k_i}{x_i}.
In Bayesian inference, the conjugate prior for the parameter \lambda of the Poisson distribution is the Gamma distribution.
\lambda \sim \Gamma(\alpha, \beta).
The Poisson parameter \lambda is distributed accordingly to the parametrized Gamma density g in terms of a shape and inverse scale parameter \alpha and \beta respectively:
g(\lambda|\alpha, \beta) = \displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha - 1} e^{-\beta \lambda}. For \lambda > 0.
Then, given the same sample of n measured values k_i from our likelihood and a prior of \Gamma(\alpha, \beta), the posterior distribution becomes: \lambda \sim \Gamma (\alpha + \displaystyle\sum_{i=1}^{\infty} k_i, \beta +n).
The posterior mean E[\lambda] approaches the maximum likelihood estimate in the limit as \alpha and \beta approach 0.
See also
References
- SOCR Home page: http://www.socr.ucla.edu
"-----
Translate this page:
