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: \[x \sim B(n,P)\]

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




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