Edit: Should I post this on Cross Validated instead?
On the Wikipedia page for Variational Bayesian methods, it is stated that
In variational inference, the posterior distribution over a set of unobserved variables $Z = \{Z_{1}\dots Z_{n}\}$ given some data $X$ is approximated by a so-called variational distribution, $Q(Z)$: $P(Z|X) \approx Q(Z)$
Here the variational distribution $Q(Z)$ is just a distribution for $Z$, not a conditional distribution of $Z|X$. However, later in the entry we have expressions such as
$$ Q(Z) = \prod_{i=1}^M q_i(Z_i|X)$$
Since
$$ q_j^*(Z_j|X) \propto E_{i \neq j} \log P(Z,X)$$
and the expectation is taken with respect to all other variables in $Z$ but $Z_j$ but not with respect to the data, I understand that $q_j^*$ should be the a conditional distribution (on the data). However, the notation $Q(Z)$ and not $Q(Z|X)$ is widespread.
Adding to my confusion is e.g. Blei et al. (2017) who explicitly write $q_j(z_j)$ (see e.g. equation 15) and note
We emphasize that the variational family is not a model of the observed data—indeed, the data x does not appear in Equation (15)
Thoughts on this? My own understanding is that contrary to posterior distributions, which are distributions for $Z|X$, variational distributions are only functions of $Z$. Is the conditioning in the Wikipedia page used to suggest that different data $X$ would result in different approximations $Q(Z)$, even though such approximations are not conditional distributions themselves?