# Poisson univariate

Suppose ${\displaystyle {\mathbf {x}}=(x_{1},x_{2},\cdots ,x_{n})}$ where ${\displaystyle x_{i}}$ are iid observations from the Poisson distribution with mass function:

${\displaystyle {\mathcal {P}}(x\mid \lambda )={\frac {e^{-\lambda }\lambda ^{x}}{x!}},\,\,x\in \{0,1,2,\ldots \},\,(\lambda >0).}$

In the parameterization above the parameter ${\displaystyle \lambda }$ represents the mean and the variance.

Overall Recommended (OR) prior
${\displaystyle \pi (\lambda )\propto \lambda ^{-1/2}}$

### Formal endorsements

The OR prior is the Jeffreys prior and also the reference prior.

### Posterior propriety

The posterior distribution of the OR prior is

${\displaystyle \pi (\lambda \mid {\mathbf {x}})={{\mathcal {G}}a}{\Big (}\lambda \mid \sum _{i=1}^{n}x_{i}+1/2,n{\Big )}}$,

where ${\displaystyle {{\mathcal {G}}a}(\alpha ,\beta )}$ represents the gamma distribution with mean ${\displaystyle \alpha /\beta }$.

### Alternatives

• A proper prior that has the OR prior as the limiting case is ${\displaystyle {{\mathcal {G}}a}(1/2,\epsilon )}$ with ${\displaystyle \epsilon \rightarrow 0}$.