In my lecture material, there is given following directed graph:
a-->c-->b
The graph can be represented by the factorization: $$p(a,b,c)=p(a)p(c|a)p(b|c)$$
The task is to check, whether $a$ and $b$ are independent, if c is unobserved. So to get $p(a,b)$ we marginalize $p(a,b,c)$ over $c$: $$p(a,b)=\sum_{c}p(a)p(c|a)p(b|c)=p(a)\sum_{c}p(c|a)p(b|c)=p(a)p(b|a)\neq p(a)p(b)$$
I tried for hours to calculate why $\sum_{c}p(c|a)p(b|c)=p(b|a)$, but I just can't get it. Is there a rule or something that I'm missing? Or is this step not as trivial as the lecture notes make it out to be?
Any help is much appreciated.