Is there a way to find $P(A | B, C)$ if I know $P(A|B) , P(A|C)$ and I also know that $B, C$ are independent

Question

So, suppose I know $P(A|B) , P(A|C)$ and I also know $P(B), P(C)$, and I also know $B, C$ are independent. From this information, is there a way to find $P(A| B, C) = \frac{P(A \cap B \cap C)}{P(B \cap C)}= \frac{P(A \cap B \cap C)}{P(B)P(C)}$. From the last equality I can't seem to figure out the numerator, and I'm starting to doubt if this is even solvable.

Thanks!

Answer 1

You can't. Suppose $B$ and $C$ are tosses of a fair coin, and $$ P(A|B, C) = a\\ P(A|B, C^c) = 1-a\\ P(A|B^c, C) = 1-a $$ Regardless of the value of $a$, you have $P(A|B)=P(A|C)=1/2$, so how could you calculate $P(A|B,C)=a$ from that?