 Solution

The joint density factors as $p(\psi ){\prod }_{j}p({\theta }_j\mid \psi )p(y_j\mid {\theta }_j)$. In the full conditional distribution of ${\theta }_i$, all factors not involving ${\theta }_i$ cancel. The remaining likelihood function and exponential distribution prior giveIn shape–rate convention this is $\boxed{{\theta }_i\mid \psi ,y\sim \mathrm{Gamma}(y_i+1,1+\psi )}$, a special case of the Gamma–Poisson hierarchical model. It depends only on the node's observed child $y_i$ and parent $\psi$, as required for its Gibbs sampler update.