Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 216 / 3 / a / Solution

Use the shape-rate convention for every gamma distribution, with $a,c>0$. Let $A_i=a+y_i$ and $s={\sum }_i{\theta }_i$. The Poisson distribution likelihood and the Gamma–Poisson hierarchical model giveAll factors involving the coordinate being updated must be retained, including $b^{na}$ from the conditional gamma distributions. The full conditional distributions areFor a blocked Gibbs sampler, draw $b$ from the second full conditional distribution, then redraw all the ${\theta }_i$ independently from the first ones. Repeating these two steps preserves the joint posterior distribution; observing $\theta$ after each complete sweep gives the Markov chain used in the next part. Standard gamma distribution sampling works for noninteger as well as integer shapes. The positive observation-period convention $t_i>0$ is used for the subsequent geometric drift condition.