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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?

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Yes you can. Suppose $Z$ can take values in $\mathcal{Z}$ and that $C$ is conditionally independent of $X$ given $Z$, i.e. $P(C=c|Z=z,X=x) = P(C=c|Z=z)$.

If $Z$ is discrete, $$ P(C=c|X=x) = \sum_{z\in \mathcal{Z}} P(C=c,Z=z|X=x) = \sum_{z\in \mathcal{Z}} P(C=c|Z=z)P(Z=z|X=x).$$ If $Z$ is continuous, $$ P(C=c|X=x) = \int_{z\in \mathcal{Z}} p(C=c,z|X=x) dz= \int_{z\in \mathcal{Z}} p(C=c|Z=z)p(z|X=x) dz.$$ The upper case $P$ indicate probability mass functions while the lower case $p$ indicate probability density functions (or a mix).

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