I have to calculate/approximate the following multiple conditional probability:
with only $P(X|C_1)$, $P(X|C_2)$ and $P(X|C_3)$ given. Is it possible to express the above in terms of these single conditional probabilities only? What independence assumptions would I need to make?
Edit: Concretely, I have an algorithm that can calculate single conditional probabilities $P(X|C_i)$ directly, and I want to use these to calculate multiple conditional probabilities. I'm fairly certain it is not possible to calculate this exactly, but perhaps I can make some independence assumptions so some terms can be dropped?