Expressing conditional independence in terms of third conditionally independent variable

Question

So we have three variables $A$, $B$, and $C$.

We know that $A$ and $B$ are conditionally independent given $C$ (ie $P(A,B|C)=P(A|C)*P(B|C)$)

How would I prove $P(C|A)=\frac{P(B|A)}{P(B|C)}$.

My first intuition was using Bayes Rule but I'm not sure where to go from there. Any hints/tips would be appreciated!

Answer 1

Bayes' rule is not in fact necessary. By the definition of conditioning, $$\mathbb{P}[B|A]=\mathbb{P}[B|C]\mathbb{P}[C|A]$$ Now divide by $\mathbb{P}[B|C]$.