# Binomial univariate

Suppose ${\displaystyle x}$ is an observation from the Binomial distribution with mass

${\displaystyle {\cal {B}}(x\mid n,\theta )={n \choose x}\theta ^{x}(1-\theta )^{n-x},\,\,x\in \{0,1,2,\ldots ,n\}\,\,(n\in \{1,2,\ldots ,\},\,\theta \in [0,1]).}$

In the parameterization above the parameter ${\textstyle \theta \,}$represents the probability of success while ${\displaystyle n\,}$represents the number of independent experiments.

## Case 1: Prior for ${\displaystyle \theta ,\,}$${\displaystyle n\,}$is known

Overall Recommended (OR) prior
${\displaystyle \pi (\theta )={\frac {1}{\pi }}\theta ^{-1/2}(1-\theta )^{-1/2}}$

### Formal endorsements

The OR prior is the reference prior (see [1]) that also coincides with the Jeffreys prior as this is a univariate parameter of a model with regularity conditions described in [1]. The OR prior is a Beta distribution with both parameters 1/2 (that is ${\displaystyle \pi (\theta )={\cal {Be}}(\theta \mid 1/2,1/2)}$)

### Posterior propriety

The OR prior is proper so is the posterior distribution. The posterior distribution has the expression:

${\displaystyle \pi (\theta \mid x)={\cal {Be}}(\theta \mid x+1/2,n-x+1/2).}$

### Alternatives

The uniform distribution ${\displaystyle {\cal {U}}(\theta \mid 0,1)\,}$is an obvious candidate for an objective prior. The posterior distribution in this case is

${\displaystyle \pi (\theta \mid x)={\cal {Be}}(\theta \mid x+1,n-x+1).}$

On the other hand, the maximal data information prior is ${\displaystyle \pi (\theta )=1.6186\theta ^{\theta }(1-\theta )^{1-\theta }\,}$but the associated posterior distribution does not have a closed form. Alternatively, it was proposed by [2] the use of the improper prior

${\displaystyle \pi (\theta )\propto \theta ^{-1}(1-\theta )^{-1}\,}$

that produces the posterior distribution ${\displaystyle {\cal {Be}}(\theta \mid x,n-x)\,}$ which is proper if ${\displaystyle 0This prior corresponds to assuming ${\displaystyle \pi (\log(\theta /(1-\theta ))\propto 1.}$

## Case 2: Priors for ${\displaystyle n,\,}$ when ${\displaystyle \theta \,}$is known.

Overall Recommended (OR) prior
${\displaystyle \pi (n)=1/{\sqrt {n}}}$

### Formal endorsements

The OR prior is the prior recommended by [3]. It is derived embedding the Binomial model in a continuous model and treating ${\displaystyle n\,}$as a continuous parameter then obtaining the reference prior. While this strategy is not unique (as there are several embedding possibilities) [3] show that the OR prior is superior as it has better frequentist performance on credible sets.

### Posterior propriety

The posterior distribution, which is not available in closed form, is proper. To prove it we need to show that the sum${\displaystyle \sum _{n=1}^{\infty }{\cal {B}}(x\mid n,\theta )\pi (n)\propto \sum _{n=1}^{\infty }{n \choose x}(1-\theta )^{n}\pi (n)}$

is convergent which follows after a straightforward application of the ratio test.

## References

1. Bernardo, J. and Smith, A. (1994), Bayesian Theory, John Wiley and Sons, London.
2. Novick, M. R. & Hall, W. J. (1965), A Bayesian indifference procedure, Journal of the American Statistical Association, 60, 1104–1117.
3. Berger, J.O, Bernardo, J.M. and Sun, D. (2012), Objective priors for discrete parameter spaces, Journal of the American Statistical Association, 107, 636-648.