Peripheralization of Conditional Distributions of head to tail model

Question

If we peripheralize the concurrent probability of the head to tail graphical model for c, we get the following equation. $p(a,b) = p(a)Σ_{c}p(a|c)p(c|b) = p(a)p(b|a)$

The question is, does the equality $Σ_{c}p(a|c)p(c|b) = p(a|b)$ hold for head to tail, and not necessarily for the general non-head to tail simultaneous distribution p(a,b,c)?

Answer 1

If we peripheralize the concurrent probability of the head to tail graphical model for c, we get the following equation. $p(a,b) = p(a)Σ_{c}p(a|c)p(c|b) = p(a)p(b|a)$

  No, we get $p(a,b)=p(a)\sum_cp(c\mid a)\,p(b\mid c)=p(a)p(b\mid a)$ .

The question is, does the equality $Σ_{c}p(a\mid c)p(c\mid b) = p(a\mid b)$ hold for head to tail, and not necessarily for the general non-head to tail simultaneous distribution p(a,b,c)?

  The equality $\sum_{c}p(c\mid a)\,p(b\mid c) = p(b\mid a)$ holds for any case where $c$ and $a$ are conditionally independent when given $b$; which means $p(c\mid a)=p(c\mid a,b)$.

  In general $\sum_c p(c\mid a,b)\,p(b\mid a)=p(b\mid a)$ always holds (by rule of total probability).