The function in question is (from Appendix B, Proof of proposition 2.1 from Posterior Regularization for Structured Latent Variable Models):
$$q(\textbf{Z}) = \frac{p_{\theta}(\textbf{Z}|\textbf{X})exp(\lambda^T \cdot \Phi(\textbf{Z}|\textbf{X}))}{H}$$
The $q(\textbf{Z})$ is a probability distribution of latent variable $z$, such that $\textbf{Z} \in \mathbb{R}^{N \times 1}$, $\textbf{X}$ is a vector of $N$ datapoints such that $\textbf{X} \in \mathbb{R}^{N \times 2}$. The $\lambda$ is a dual variable, such that $\lambda \in \mathbb{R}^{N \times 1}$ and is a vector function $\Phi(\textbf{Z}|\textbf{X})$. $H$ is a constant that normalizes the distribution to have a sum of $1$.
The question is, how to evaluate $q()$ for a single variable $z$, rather than a vector $\textbf{Z}$ ?
Particular example of confusion: For instance, for particular $z_i$, which is related to datapoint $x_j$, is the term in exponential this $exp(\lambda^T \cdot \Phi(\textbf{Z}|\textbf{X}) )$ or this $exp(\lambda^T_i \cdot \Phi(z_i|x_j) )$?
In other words, is a single $z$ weighted through the exponential term obtained from all other $\textbf{Z}$, or just the particular $z_i$?
If its the former case, do I need to explicitly define $\Phi(z_k|x_j) = 0$, when $i \ne k$, because otherwise it seems undefined as a vector function for the elements $\Phi()$ not defined for $z_i$.
General question: How to intuitively understand whether my distribution/function is a function of only particular variable or a vector of variables?