= Solution
The joint density factors as $p(\psi)\prod_jp(\theta_j\mid\psi)p(y_j\mid\theta_j)$. In the <full conditional distribution> of $\theta_i$, all factors not involving $\theta_i$ cancel. The remaining <likelihood function> and <exponential distribution> prior give
$$
p(\theta_i\mid\theta_{-i},\psi,y)\propto\theta_i^{y_i}e^{-\theta_i}e^{-\psi\theta_i}
=\theta_i^{y_i}e^{-(1+\psi)\theta_i}.
$$
In shape–rate convention this is $\boxed{\theta_i\mid\psi,y\sim\operatorname{Gamma}(y_i+1,\,1+\psi)}$, 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.
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