For a family of distribution [mathjaxinline]\, \{ \textsf{Poiss}(\lambda )\} _{\lambda >0} \,[/mathjaxinline] , Jeffreys prior is proportional to:
[mathjaxinline]\, \pi _ j(\lambda ) \propto \,[/mathjaxinline]
[mathjaxinline]\, \sqrt{I(\lambda)} \propto \sqrt{E\left[ \left( \frac{\partial \log f(X|\lambda)}{\partial \lambda} \right)^2 \right]} \propto \sqrt{E\left[ \left( \frac{\partial \log \left( \frac{e^{-\lambda}\lambda^X}{X!} \right)}{\partial \lambda} \right)^2 \right]} \propto \sqrt{E\left[ \left( -X\frac{e^{-\lambda}\lambda^{X-1}}{\lambda^X} \right)^2 \right]} \propto \sqrt{E\left[ \left( -\frac{X}{\lambda} \right)^2 \right]} \propto \sqrt{E\left[ \frac{X^2}{\lambda^2} \right]} \propto \sqrt{\frac{E[X^2]}{\lambda^2}} \propto \frac{\sqrt{\lambda}}{\lambda} = \frac{1}{\sqrt{\lambda}}[/mathjaxinline]
The Jeffreys prior for a family of distributions is proportional to the square root of the Fisher information. To find the Jeffreys prior for the Poisson distribution, we need to calculate the Fisher information for the Poisson distribution.
The Fisher information for the Poisson distribution is given by:
[mathjaxinline]\, I(\lambda ) = \frac{1}{\lambda } \, .[/mathjaxinline]
Therefore, the Jeffreys prior for the Poisson distribution is proportional to the square root of the Fisher information:
[mathjaxinline]\, \pi _ j(\lambda ) \propto \sqrt{I(\lambda )} = \sqrt{\frac{1}{\lambda }} \, .[/mathjaxinline]