## FANDOM

10 Pages

In this section $x_1,x_2,\cdots,x_n$ are iid observations from the Binomial distribution with density:

$f(x\mid p,n)\propto p^{x} (1-p)^{n-x} ,\,0<x<n,\,(0<p<1,n\in \mathbb{Z}^{+}).$

In the parameterization above $p$ represent the probability of an exitus while $n$ is the number of repetitions of the experiment and hence, the maximum value for $x$.

## Case 1: p unknown, n known Edit

Case 1: Recommended prior
$\pi(p \mid n) = Beta(1/2,\, 1/2)$

### Formal endorsements Edit

This Beta distribution was first proposed by [1] and is the reference as well as the Jeffreys prior for $p$.

### Posterior propriety Edit

The posterior distribution associated with this prior is always proper given that the prior it is so. In particular, the (marginal) posterior is

$Beta(p\mid x+1/2,\,n-x+1/2)$

### Alternatives Edit

[2] propose an improper prior also known as the Haldane prior.

$\pi(p\mid n)\propto p^{-1} (1-p)^{-1}.$

This is a limiting case of a $Beta(k,k)$ when $k\rightarrow 0$

## Case 2: Prior for n Edit

Case 2: Recommended prior
$\pi(n)\propto 1/n$

### Formal endorsements Edit

This prior is the reference prior discussed by [3].