How can q(z,l,u,c|x) be factorized to q(z|x)q(l|x)q(c|z)q(u|c,z)? How can q(z,l,u,c|x) be factorized to q(z|x)q(l|x)q(c|z)q(u|c,z)?
I don't know what assumptions are behind this equation, which is I want to know. I think normal chain rule of probability is in the process, but only chain rule can't derive the formula.
What I know is that z, l, u follows gaussian distribution, and c follows multinomial distribution.
It's from the paper Xu C, Lopez R, Mehlman E, Regier J, Jordan MI, Yosef N. Probabilistic harmonization and annotation of single-cell transcriptomics data with deep generative models. Mol Syst Biol. 2021 Jan;17(1):e9620. doi: 10.15252/msb.20209620. PMID: 33491336; PMCID: PMC7829634.
That formula is used for derive ELBO, since the model is similar to VAE.

It is a basic theorem of conditional probability that $$P(A,B \mid C) = P(A \mid B,C) \cdot P(B \mid C).$$
Furthermore, if $A$ and $B$ are conditionally independent given $C$, then $P(A \mid B,C) = P(A \mid C)$, and so $$P(A,B \mid C) = P(A \mid C) \cdot P(B \mid C).$$
Conversely, if $A$ depends on $C$ only via $B$, then $P(A \mid B,C) = P(A \mid B)$, and so $$P(A,B \mid C) = P(A \mid B) \cdot P(B \mid C).$$
The factorization $q(z,l,u,c \mid x) = q(z \mid x)q(l \mid x)q(c \mid z)q(u \mid c,z)$ given in your question can thus be interpreted as a description of how the various variables are assumed to (approximately) depend on each other.  In particular, we can see that:

*

*$z$ and $l$ are assumed to be conditionally independent, given $x$; and

*$c$ and $u$ are assumed to depend on $x$ only via $z$ (and to be conditionally independent of $l$).

(Note that we cannot say anything about the dependence between $c$ and $u$, except that it is assumed to be potentially non-negligible.  In particular, $q(c \mid z) q(u \mid c,z)$ and $q(c \mid u,z) q(u \mid z)$ are both equally valid factorizations of $q(c,u \mid z)$.  Remember that statistical dependence is not causality, and always goes both ways, even when the notation might make it look otherwise.)
