I'm rather confused on how to perform variable elimination in the following bayesian believe network.
If I have the following query queryP(B|G, E) with a variable ordering of G, E, A, B, C, D, F.
My approach:
P(A,B,C,D,E,F,G) = P(A) * P(B|C|D|E|F, A) * P(C|G, B) * P(D|G, C) * P(E|D) * P(F|E)
Giving me {f0(A), f1(A,B,C,D,E,F), f2(C,G,B), f3(D,G,C), f4(E,D), f5(F,E)}
I condition on G and E being true then sum out A,B,C,D,F. Leaving me with an empty set. I don't believe this to be correct and am fundamentally misunderstanding something on the elimination process.
Hint: The joint probability factored according to the network is $$P(A,\ldots,G) =P(A)P(B|A)P(C|A)P(D|A)P(E|A)P(F|A)P(G|B,C).$$ Then $P(B,E,G)$ is the marginal probability of $P(A,\ldots,G)$, where you sum over all remaining variables. Similar for $P(E,G)$. Then you can use the definition of conditional probability.