 Solution

Multiplying the Poisson distribution likelihood by the Jeffreys prior gives the posterior density kernelThus, in shape-rate convention,It is a proper posterior distribution even when $y_i=0$, provided $E_i>0$. The posterior mean and posterior variance are $(y_i+1/2)/E_i$ and $(y_i+1/2)/E_i^2$. This is Poisson-gamma conjugacy with unequal exposures, with the zero-rate prior interpreted as an improper kernel.