Express multiple conditional probability in terms of single conditional probabilities I have to calculate/approximate the following multiple conditional probability:

$$P(X|C_1,C_2,C_3)$$

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?
Assume $C_1,C_2,C_3$ are independent of each other. Then

$$P(X|C_1)P(X|C_2)P(X|C_3)=\frac{P(XC_1)}{P(C_1)}\cdot\frac{P(XC_2)}{P(C_2)}\cdot\frac{P(XC_3)}{P(C_3)}\\ =\frac{P(XC_1C_2C_3)}{P(C_1C_2C_3)}=P(X|C_1C_2C_3)$$
Assume $C_1,C_2,C_3$ are independent of each other. Then

$$P(X|C_1)P(X|C_2)P(X|C_3)=\frac{P(XC_1)}{P(C_1)}\cdot\frac{P(XC_2)}{P(C_2)}\cdot\frac{P(XC_3)}{P(C_3)}\\ =\frac{P(XC_1C_2C_3)}{P(C_1C_2C_3)}=P(X|C_1C_2C_3)$$