Reversing parenthood

So say we had this graph with $p(c | b), p(b | a), p(a)$. Can we find $p(B | A, C)$?

Actually, it’s pretty straightforward. You want to go with the flow, meaning that you want to find $P(C, B | A)$ first. Then, to flip it, just divide by $P(C | A)$ because according to bayes rule, it gets you $P(B | C, A)$.

Let’s start with $P(C, B | A)$. Due to conditional independence, it’s just $P(B | A)P(C | B)$, which you have.

Now, to get $P(C | A)$, just marginalize the $B$!