Having $x, y, z, c \in \mathbb{R}$, is it valid to say:
$c \propto g(x, z) h(y, z)$
The context here is to say whether or not the random variables $X$ and $Y$ are independent given the value of $Z$.
The above statement is one of the series of four statements that validates $P \models X \bot Y \mid Z$, having $c$ as the expression.
For example, if I had $cxyz$ as the expression, $cxyz \propto g(x, z) h(y, z)$ is valid, since I'm able to do $g(x, z) = cxz;\ h(y, z) = cyz$. I'm not sure if I can say $g(x, z) = c;\ h(y, z) = c$, though.
Is it valid?
Yes, that implies conditional independence.
Write the conditional density and show that it factors into a product
\begin{eqnarray*} f \left( x, y|z \right) & = & \frac{f \left( x, y, z \right)}{f \left( z \right)}\\ & \propto & \underbrace{g \left( x, z \right)}_{\text{depends only on } x} \times \underbrace{h \left( y, z \right)}_{\text{depends only on } y}\\ & = & f \left( x|z \right) f \left( y|z \right) \end{eqnarray*}
Obviously the proportionality constant in the last equality is different from the one in the definition of the joint distribution (it could depend on z).