In page 21 here, it states:
General Idea of Amortization: if same inference problem needs to be solved many times, can we parameterize a neural network to solve it?
Our case: for all $x^{(i)}$ we want to solve: $$ \min _{q(z)} \mathrm{KL}\left(q(z) \| p_{\theta}\left(z \mid x^{(i)}\right)\right. $$ Amortized formulation: $$ \min _{\phi} \sum \operatorname{KL}\left(q_{\phi}\left(z \mid x^{(i)}\right) \| p_{\theta}\left(z \mid x^{(i)}\right)\right) $$
One thing I am trying to wrap my mind around is $q(z)$ in the 1st formulation vs $q_{\phi}(z \mid x^{(i)})$ in the second.
Why is one a conditioned on $x^{(i)}$ and the other is not? I know in the 1st formulation, we are trying to find a different $q(z)$ for each datapoint.
I also recall that in VAEs,which uses amortized inference we consider $q(z)$ to be aggregated posterior, like
$$q_{\phi}(z)=\int q_{\phi}(x, z) d x \quad \text { Marginal of } q_{\phi}(x, z) \text { on } z$$
$$q_{\phi}(x, z) \equiv p_{\mathcal{D}}(x) q_{\phi}(z \mid x)$$
(formulas taken from here)
