Trying to calculate the $P(A|\bar{C})$ based on below BBN and wondering if someone can help me confirm the correct setup from below approaches.
Approach 1:
$P(A|\bar{C}) = \sum_{B,D,E} P(B|\bar{C})P(D|\bar{C})P(E|\bar{C})P(A|B)$
I feel like this approach is a non-starter because ($P(B|\bar{C})P(D|\bar{C})P(E|\bar{C})$) implies B,D and E are conditionally independent given C which according to BBN is wrong.
Approach 2:
$P(A|\bar{C}) = \frac{P(\bar{C}|A) P(A)}{P(\bar{C})}$
then $ P(\bar{C}|A) = \sum_{B,D,E} P(\bar{C}|B,D,E)P(A)P(B|A) $
Approach 3:
$P(A|\bar{C}) = \sum_{B,D,E} P(B,D,E | \bar{C})P(A|B) $