AP Statistics Curriculum 2007 Bayesian Other
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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
- , (x = 0, 1, …, n)
- p(x|P) is proportional to P^{x}(1 − P)^{n − x}.
If the prior density has the form:
- , (P between 0 and 1),
then it follows the beta distribution
- .
From this we can appropriate the posterior which evidently has the form:
- .
The posterior distribution of the Binomial is
- Failed to parse (lexing error): (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 λ if it has the density:
Suppose that you have n observations from such a distribution so that the likelihood is:
- L(λ | x) = λ^{T}e^{( − nλ)}, where .
In Bayesian inference, the conjugate prior for the parameter λ of the Poisson distribution is the Gamma distribution.
- .
The Poisson parameter λ is distributed accordingly to the parametrized Gamma density g in terms of a shape and inverse scale parameter α and β respectively:
- . For λ > 0.
Then, given the same sample of n measured values k_{i} from our likelihood and a prior of Γ(α,β), the posterior distribution becomes:
- .
The posterior mean E[λ] approaches the maximum likelihood estimate in the limit as α and β approach 0.
See also
References
- SOCR Home page: http://www.socr.ucla.edu
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