Graphic model factorizing, marginalization

Question

This is actually a probability marginalization question that I encountered in graphic models section of PRML by Bishop (question about equation 8.26 page 391). Assume I have the following graphic model

Therefore the joint probability density of the variables will factor to $p(a,b,c)=p(a)p(c|a)p(b|c)$. Now assume I want to marginalize over $c$, the book says

$$\sum_c p(a,b,c) = \sum_c p(a)p(c|a)p(b|c) = p(a) \sum_c p(c|a)p(b|c)= p(a)p(b|a)$$

This means that $p(b|a) = \sum_c p(c|a)p(b|c)$, how to prove this?! If it was like this $\sum_c p(c|a)p(b|c,a)$ then one can reason that $\sum_c p(c|a)p(b|c,a) = \sum_c \frac{p(a,c)}{p(a)}\frac{p(a,b,c)}{p(a,c)} = \sum_c p(b,c|a)=p(b|a)$. But I can't conclude the same result with $\sum_c p(c|a)p(b|c)$. What am I getting wrong?

Thanks in advance

Answer 1

You need to use the fact that $p(b|c,a)=p(b|c)$ i.e. $a$ and $b$ conditionally independent given $c$. This is directly visible on the graph and also proved with:

$p(b|c,a)=\frac{p(b,c,a)}{p(c,a)}=\frac{p(a)p(c|a)p(b|c)}{\sum_{b}p(a)p(c|a)p(b|c)}=\frac{p(b|c)}{\sum_{b}p(b|c)}=p(b|c).$

Use this additional equality at the beginning of your development to get the result.