Assume we have such a simple graphical directed model of probability. As a consequence we can factorize joint probability as following: $$p(x,y,z)=p(y|x,z)p(x|z)p(z)=p(y|x,z)p(x)p(z)$$
In directed graphical model every random variable can be written in the conditional (in)dependence context. My question is following: whether $p(x|z)=p(x)$ is a consequence of conditional independence rule or independence rule? Independence of variables implies conditional independence?
1) In case of conditional independence, can we say that: $$x \perp z,y |0 $$ so as a result $$p(x|0,z,y)=p(x|0)=p(x)$$ ?? Can we even condition on zero?
2) In case of not conditional independence,things are much more clear: $$p(x|z)=\frac{p(x)p(z)}{p(z)}=p(x)$$
3)Here is some kind of appendix for question 1) (conditional independence of given graphical directed model):
