 Solution

Use the standard interpretation that the counts are conditionally independent given the parameters, with mutually independent stated priors. Let $a_2={\alpha }_2$, $b_2={\beta }_2$, and defineUnder the reference constraints, the Poisson distribution likelihood, up to data-only factors, isMultiplying by the shape-rate gamma prior gives the Poisson-gamma conjugacy with unequal exposuresHere $b$ is a rate, matching the probability density function convention on the PDF's first page. The factor $I$ represents the number of replicates per cell, not the total count.