In the recent paper by Justin Gilmer, the following identity is assumed to hold: given finite random variables $X, X' \in \{0,1\}; C, C' \in \{1,..,M\}$, such that the pairs $(X, X')$, $(C, C')$, $(X, C')$, $(X', C)$ - but not $(X, C)$, $(X', C')$ - are independent, we have:
$$P(X = 0, X' = 0 | C = c, C' = c') = P(X = 0 | C = c) * P(X' = 0 | C' = c')$$ Specifically, in Lemma 1 (the restated version, page 4) and Lemma 5 (the first equation of the proof, page 8). My questions are:
- Am I correct in assuming that this identity is necessary for the derivation of said resutlts?
- Are the given independence conditions enough for this identity to hold true?
EDIT: I should've reversed the order of the questions - question 2 is the main question I'm looking to get an answer to.
EDIT 2: it is also known that pairs $(X,C)$ and $(X',C')$ are identically distributed, specifically: $$P(C=c) = P(C'=c)=q(c),\ P(X=x|C=c) = P(X'=x|C'=c) = p_c$$