This should be really simple but feels like I'm missing something.
Consider that we have random variables that influence each other as per $z \rightarrow x \rightarrow y$. Then $y$ is independent of $z$ conditional on $x$. Namely we get:
$$\begin{align} p(y|x, z) &= p(y|x) \\ \frac{p(y, x, z)}{p(x, z)} &= \frac{p(y, x)}{p(x)} \end{align} $$ if we integrate out $x$ from the two sides we get: $$\begin{align} \frac{p(y, z)}{p(z)} &= p(y)=p(y|z) \end{align} $$ which seems to suggest that $y$ and $z$ are independent of each other, which doesn't make sense as surely knowing the value of $z$ tells us something about the distribution of $y$. Am I doing something wrong with my integrating out step?