# AP Statistics Curriculum 2007 Bayesian Other

## 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.