How do we derive the conditional distribution for a Poisson whose rate is the product of two Gamma distributed rv?

Question

This question is motivated by Gopalan et al. "Content-based recommendations with poisson factorization." Advances in neural information processing systems 27 (2014). https://proceedings.neurips.cc/paper/2014/file/97d0145823aeb8ed80617be62e08bdcc-Paper.pdf. Their model can be simplified to:

1. $\alpha_{ik} \sim \text{Gamma}(a,b)$.
2. $\delta_{jk} \sim \text{Gamma}(c,d)$.
3. $z_{ijk} \sim \text{Poisson}(\alpha_{ik}\delta_{jk})$.

$i, j, \text{and } k$ are indices. Thus, the rate of the Poisson is the product of two Gamma distributed random variables. The paper gives the following conditional distribution (see Table 1):

$P \left( \alpha_{ik} \mid \delta_{jk}, z_{ijk} \right) =\text{Gamma} \left(a + \sum_j{z_{ijk}}, b + \sum_j{\delta_{jk}} \right)$.

I am familiar with conjugacy in the standard Gamma-Poisson model. I am not sure, however, of how to derive the above conditional.

Please note that the above equations are not identical to those in the paper. The model in the paper is more complex. I simplified for readability. Hence, to reduce confusion, I changed the notation. These equations, however, should track the key elements of the model and match the conditional for $\beta_{vk}$ in Table 1 in the paper.

Answer 1

Let $z_{ik} = \{z_{ijk} \forall j\}$.

Likelihood:

$P\left(z_{ik} \mid \alpha_{ik}, \delta_{jk}\right) = \prod_{j} \frac{\left(\alpha_{ik}\delta_{jk}\right)^{z_{ijk}}e^{-\alpha_{ik}\delta_{jk}}}{z_{ijk}!}$.

Prior:

$P\left(\alpha_{ik}\right) \propto \alpha_{ik}^{a-1}e^{-b\alpha_{ik}}$.

Posterior:

$P\left(\alpha_{ik} \mid z_{ik}, \delta_{jk}\right) \propto \alpha_{ik}^{a-1} e^{-b\alpha_{ik}} \prod_{j} \frac{\left(\alpha_{ik}\delta_{jk}\right)^{z_{ijk}}e^{-\alpha_{ik}\delta_{jk}}}{z_{ijk}!}$.

$P\left(\alpha_{ik} \mid z_{ik}, \delta_{jk}\right) \propto \alpha_{ik}^{a-1} e^{-b\alpha_{ik}} \prod_{j} \alpha_{ik}^{z_{ijk}}e^{-\alpha_{ik}\delta_{jk}}$.

$P\left(\alpha_{ik} \mid z_{ik}, \delta_{jk}\right) \propto \alpha_{ik}^{a+\sum_j{z_{ijk}}-1}e^{-\alpha_{ik}(b+\sum_j{\delta_{jk}})}$.

Hence: $P\left(\alpha_{ik} \mid z_{ik}, \delta_{jk}\right)$ is $\text{Gamma}(a+\sum_j{z_{ijk}}, b+\sum_j{\delta_{jk}}) $.