C given A in a Bayesian network where $A\to B\to C$?

Question

Let's say I have a Bayesian network with

A-->B-->C where  A,B,C have a Bernouilli distribution

How do I calculate $P(C=1|A=1)$? Is it $P(C=1|B=1∩A=1) + (C=1|B=0∩A=1)$ or $P(C=1∩B=1|A=1) + (C=1∩B=0|A=1)$ or something else?

Thanks

Answer 1

In general it is $$P(C=1\mid A=1) \\=P(C=1\cap B=1\mid A=1) + P(C=1\cap B=0\mid A=1)$$ which you can extend to $$ = P(C=1\mid B=1\cap A=1)P(B=1\mid A=1) \\+ P(C=1\mid B=0\cap A=1)P(B=0\mid A=1)$$

but if your particular model of the relationship is correct, this suggests you could go further and say

$$ = P(C=1\mid B=1)P(B=1\mid A=1) + P(C=1\mid B=0)P(B=0\mid A=1)$$