For three random variables, $X$, $Y$, and $Z$, and they all have probability densities. We say that $X$ and $Y$ are considered conditionally independent given $Z$, $(X\perp Y|Z)$ if $$ p(y|x,z)=p(y|z) $$ Now I have a question if $X\perp Y|Z$, then we have $$ \begin{aligned} p\left( y|x \right) &=\int{p\left( y|x,z \right) p\left( x|z \right) dz}\\ &=\int{p\left( y|z \right) p\left( x|z \right) dz}\\ \end{aligned} $$
So is the reverse true? That is, if $p\left( y|x \right)=\int{p\left( y|z \right) p\left( x|z \right) dz}$ holds, can we deduce that $X\perp Y|Z$? I intuitively think that it is not true. If not, are there any counterexamples?
Besides, if we want to deduce independence, what other conditions need to be added? It would be even better if there were necessary and sufficient conditions for both sides of this problem to be true.
Any suggestion is welcomed.
