How can I prove the equivalence of these two conditional independence definitions?
Definition 1: $p(x,y|z) = p(x|z) * p(y|z)$
Definition 2: $p(x|y,z) = p(x|z)$
I tried substituting Definition 2 into Definition 1: $$p(x,y|z) = p(x|y,z) * p(y|z)\\ = p(x|y) * p(x|z) * p(y|z)\\ = p(x|y) * p(y|z) * p(x|z)$$
However, I am stuck after this step. I may be wrong from the first step.
I would suggesting using the definition of conditional probability (or density):
$$p(u|v):=\frac{p(u,v)}{p(v)}.$$
So to show definition 2 implies definition 1, we have
$$p(x,y|z)=\frac{p(x,y,z)}{p(z)}=\frac{p(x|y,z)p(y,z)}{p(z)}=p(x|y,z)p(y|z)=p(x|z)p(y|z),$$
where the last step uses definition 2. Now try the other direction.