I am attempting to classify data $x$ into one of two classes $C$. I have no labelled training data but know the conditional distribution of $C$ given a proxy variable $z$, $P(C|z)$, and the distribution $P(z|x)$. I am attempting to calculate $P(C|x)$ but I can't get through the maths.
Can I infer $P(C|x)$ from the two known distributions, or do I need more information?