= Solution
Take the exposure $E_i$ as known and positive. The <Poisson distribution> <log-likelihood>, up to a parameter-independent constant, is $\ell_i(\lambda_i)=y_i\log\lambda_i-E_i\lambda_i$. Its first two derivatives are
$$
\ell_i'(\lambda_i)=\frac{y_i}{\lambda_i}-E_i,\qquad \ell_i''(\lambda_i)=-\frac{y_i}{\lambda_i^2}.
$$
Since $\mathbb E(Y_i)=E_i\lambda_i$, the <Fisher information> is
$$
\mathcal I_i(\lambda_i)=-\mathbb E\ell_i''(\lambda_i)=\frac{E_i}{\lambda_i}.
$$
The <Jeffreys prior> is proportional to its square root. Removing the constant $\sqrt{E_i}$ gives
$$
\boxed{\pi_J(\lambda_i)\propto\lambda_i^{-1/2},\qquad\lambda_i>0.}
$$
This is an <improper prior>: its integral diverges at infinity.
Back to article page
