# Gibbs sampling with Poisson Gamma models

I am trying to obtain a Gibbs sampler for a Poisson-Gamma topic model. Essentially, for each document $$d$$, the likelihood of $$d$$ depends on a Poisson parameter $$\lambda_d = \sum_k \pi_{k,d}\phi_{k,w}$$. In turn, a Gamma prior is assigned to $$\pi_{k,d}$$ (the strength of topic $$k$$ in $$d$$). Likewise, a Gamma prior is assigned to $$\phi_{k,w}$$ (the frequency of word $$w$$ in topic $$k$$).

Can you please show me how to derive a Gibbs sampler for the above model? Is it possible to conveniently collapse?

• To write a Gibbs sampler, the first step would typically be to write out the likelihood of the model, and then to derive the full conditionals by identifying conditional independence. Once you have derived full conditionals, you can then identify whether any of these have closed form. What have you tried so far? Feb 26, 2021 at 17:43
• @ Xi'an. Can you further elaborate on the introduction of the additional variables? Feb 26, 2021 at 20:00
• The likelihood of a document $d$ is $P(d) = \prod_{w \in d} (\frac{1}{n_{w,d}!} \sum_k \pi_{k,d}\phi_{k,w})^{n_{w,d}} \cdot e^{-(\sum_k \pi_{k,d}\phi_{k,w})}$, where $n_{w,d}$ is the occurrences of word $w$ in document $d$. Feb 26, 2021 at 20:16
• How to derive the full conditionals for $\pi_{k,d}$ and $\phi_{\phi_{k,w}}$. It is not clear whether the addition of further random variables is necessary and, in such a case, how to do it. Feb 26, 2021 at 20:18

## 1 Answer

Most surprisingly, this question and its resolution appear in a question you asked five years ago, in connection with a paper of Gopalan et al. (2015). As I have written my answer before realising this duplicate existed, I'll keep it posted (if only because the original post found no answer at the time).

If$$N_{w,d}\sim\mathcal P(\sum_{1\le k\le K} \pi_{k,d}\phi_{k,w})\tag{1}$$ the likelihood writes as $$\prod_d\prod_{w \in d} \frac{1}{n_{w,d}!} (\textstyle \sum_{1\le k\le K} \pi_{k,d}\phi_{k,w})^{n_{w,d}} \cdot e^{-\sum_{1\le k\le K} \pi_{k,d}\phi_{k,w}}$$ and the summation in each term creates a difficulty when simulating the parameters.

A way to avoid the summation is to create latent variables, using the property that a sum of independent Poisson variates is a Poisson variate. If we introduce the latent (unobserved) independent variables $$N_{w,d,k}\sim P(\pi_{k,d}\phi_{k,w})$$ and condition on the fact that $$N_{w,d}=\sum_{1\le k\le K} N_{w,d,k}$$ is their observed sum, then $$N_{w,k}$$ is distributed as in (1), hence the distribution of $$N_{w,d}$$ is the marginal of the joint distribution on the $$N_{w,d,k}$$'s $$\prod_d\prod_{w \in d}\prod_{k=1}^K \frac{1}{n_{w,d,k}!} [\pi_{k,d}\phi_{k,w}]^{n_{w,d,k}} \, e^{-\pi_{k,d}\phi_{k,w}} \mathbb I_{x_{w,d,1}+\cdots+x_{w,d,K}=x_{w,d}}$$ Since the $$N_{w,d,k}$$'s are not observed, they need be simulated as well in the Gibbs sampler. The simulation conditional on $$N_{w,d}$$ is however straightforward as the conditional distribution of $$(N_{w,d,1},\ldots,N_{w,d,K})$$ given their sum $$N_{w,g}$$ is a Multinomial distribution with probability vector $$(\pi_{1,d}\phi_{k,w},\ldots,\pi_{K,d}\phi_{k,w})\Big/\textstyle \sum_{1\le k\le K} \pi_{k,d}\phi_{k,w}$$ The other components of the Gibbs sampler then turn out to be straightforward as well since $$p(\pi_{k,d}) \propto \pi_{k,d}^{\textstyle\sum_{w\in d} n_{w,d,k}}\,e^{-\pi_{k,d}\textstyle\sum_{w\in d}\phi_{k,w}}\,p_0(\pi_{k,d})$$ and $$p(\phi_{k,w})\propto \phi_{k,w}^{\textstyle\sum_{d\ni w} n_{w,d,k}}\,e^{-\phi_{k,w}\textstyle\sum_{d\ni w}\pi_{k,d}}\,p_0(\phi_{k,w})$$ if $$p_0$$ denotes the appropriate prior pdf and $$p$$ the full conditional posterior pdf.

In the case when the priors are Poisson-conjugate, i.e. when$$\pi_{k,d}\sim \mathcal G(\alpha,\beta) \quad\phi_{w,k}\sim\mathcal G(\gamma,\delta)$$ $$p(\phi_{k,w})$$ corresponds to a $$\mathcal G\left(\gamma+\sum_{d\ni w} n_{w,d,k},\delta+\sum_{d\ni w}\pi_{k,d}\right)$$density and $$p(\pi_{k,d})$$ to a $$\mathcal G\left(\alpha+\sum_{w\in d} n_{w,d,k},\beta+\sum_{w\in d}\phi_{k,w}\right)$$density.

• The original model is: $\pi_{k,d} \sim Gamma(\cdot|\alpha,\beta)$; $\phi_{k,w} \sim Gamma(\cdot|\gamma,\delta)$; $N_{w,d} \sim Poisson(\cdot| \sum_{k} \pi_{k,d} \phi_{k,w} )$. Feb 27, 2021 at 14:17
• After the introduction of the additional random variables, the model becomes: $\pi_{k,d} \sim Gamma(\cdot | \alpha, \beta)$; $\phi_{k,w} \sim Gamma(\cdot|\gamma,\delta)$; $N_{w,d,k} \sim Poisson(\cdot|\pi_{k,d} \phi_{k,w})$; $N_{w,d} = \sum_k N_{w,d,k}$ Feb 27, 2021 at 14:22
• A number of points are not clear to me in the last model. Feb 27, 2021 at 14:24
• I) According to the theorem on the additivity of Poisson random variables, the added variables $N_{w,d,k}$ must be independent. In the model resulting after the introduction of the added random variables, any two such variables $N_{w,d,k_i}$ and $N_{w,d,k_j}$ appear to be conditional dependent given their child $N_{w,d}$ (which is observed, being a known amount). Can you please shed light on this point? How can the independence of the added random variables can be formally proven? Feb 27, 2021 at 14:33
• II) Also, in the context of a graphical model, should not the above notion of independence sound be connected to conditional independence or d-separation? Feb 27, 2021 at 14:34