TL;DR
In the HSLDA paper by Perotte et al, the posterior conditional distribution of $z_{d,n}$ for Gibbs Sampling is specified as:
\begin{equation*} p(z_{d,n}=k| \mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta)\propto \underbrace{\frac{n_{-i, (k)}^{w_i}+\beta}{n_{-i, (k)} + V\beta} \cdot \frac{n_{-i, (d)}^k + \alpha}{n_{-i, (d)} + K\alpha}}_{=p(z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \alpha, \beta)\text{ (not the problem)}} \cdot \underbrace{\prod_{l \in \mathcal{L}_d} \exp \left\{ -\frac{1}{2}(\bar{\mathbf{z}}_d^T\eta_l - a_{l,d})^2 \right\}}_{ = p(\mathbf{a} | \mathbf{z}, \eta)} \end{equation*}
Isn't the final term constant w.r.t. the word-assignment $z_{d,n} = k$, so we should actually be able to ignore it, right?
Background info on HSLDA:
I'm working through Hierarchically Supervised Latent Dirichlet Allocation by Perotte et al (2011), which is an extension of Blei's LDA. I am having issues understanding the update of the posterior of a the conditional distribution for the Gibbs Sampling procedure. I would like to avoid summarising the entire paper and I'll try to limit this question to only the relevant parts of the paper. Further below, you will find the graphical model and the generative model.
Perotte et al start describing that HSLDA extends LDA in that it allows for $K$ latent topics and document labels. The incorporation of hierarchical document labels is what sets HSLDA apart from other flavours of LDA. The two parts of the models are connected via the word-topic assignments $z_{d,n}$.
The practical use I see in HSLDA, is that it allows to find distinguishing features between document labels that are inherently close to each other, because they are in the same branch in the document label tree.
Deriving the conditional posterior of $z_{d,n}$
is the conditional distribution of the topic-assignment $z_{d,n}$ required to draw samples from the HSLDA posterior distribution using Gibbs sampling and Markov chain Monte Carlo. I have experienced a couple of times now that computer science papers are extremely brief and usually simply state results, rather than derivations. Equation1 in the paper can be derived as follows:
\begin{align*} p(z_{d,n}=k | \mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta) &= \frac{p(z_{d,n}, \mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta)}{p(\mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta)}\\ &=\frac{p(\mathbf{a}, z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \eta, \alpha, \beta)\cdot p(\mathbf{z}_{-d,n}, \mathbf{w}, \eta, \alpha, \beta)}{p(\mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta)}\\ &\propto p(\mathbf{a}, z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \eta, \alpha, \beta) \\ &= p(\mathbf{a} | z_{d,n}, \mathbf{z}_{-d,n}, \mathbf{w}, \eta, \alpha, \beta)\cdot p(z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \eta, \alpha, \beta) \\ &=p(\mathbf{a} | \mathbf{z}, \eta) \cdot p(z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \alpha, \beta) \end{align*}
By the generative model specification, this results in:
\begin{equation*} p(z_{d,n}=k| \mathbf{z}_{-d,n}, \mathbf{a}, \mathbf{w}, \eta, \alpha, \beta)\propto \underbrace{\frac{n_{-i, (k)}^{w_i}+\beta}{n_{-i, (k)} + V\beta} \cdot \frac{n_{-i, (d)}^k + \alpha}{n_{-i, (d)} + K\alpha}}_{=p(z_{d,n} | \mathbf{z}_{-d,n}, \mathbf{w}, \alpha, \beta)\text{ (not the problem)}} \cdot \underbrace{\prod_{l \in \mathcal{L}_d} \exp \left\{ -\frac{1}{2}(\bar{\mathbf{z}}_d^T\eta_l - a_{l,d})^2 \right\}}_{ = p(\mathbf{a} | \mathbf{z}, \eta)} \end{equation*}
My question
The result is the posterior conditional distribution that we use to sample the values for $z_{d,n} \in \{1,2,...,K\}$ from. Perotte et al keep the last term $p(\mathbf{a} | \mathbf{z}, \eta)$ in the update. However, that term only affects the posterior conditional distribution by a multiplicative constant, because $\bar{\mathbf{z}}_d^T \eta_l$ does not change with $k$ and neither does $a_{l,d}$. Since the sum $\sum_{k=1}^K p(z_{d,n} =k | ...)$ is normalised, it actually does not have any affect at all.
Somehow I feel there should be a term in that posterior conditional distribution, that involves the "bottom part" of the graphical model, but as I described above, this specific term doesn't make sense to me, really. What is the reason that Perotte left that term in the posterior update?
