Is there a way to estimate $\mathbb P(X|A,B)$ given ONLY an estimate for $\mathbb P(X|A)$ and $\mathbb P(X|B)$? if not, what other information is needed? Is it possible if I assume $A$ and $B$ are independent?
Furthermore, given an estimate for $\mathbb P(X|A_1\cap A_2\cap \cdots\cap A_n)$, how do I find an estimate for $\mathbb P(X|A_1\cap A_2\cap \cdots\cap A_n\cap A_{n+1})$? Can I get this estimate if I know $\mathbb P(X|A_{n+1})$? What other information do I need? Does it help if all $A_i$ are independent?
Here is an example showing there is little hope to reach a positive statement.
Let the random variables $(U,V)$ be uniform on $\{0,1\}^2$, $A=[U=1]$, $B=[V=1]$, $X=A\cap B$, and $Y=A\setminus B$. Then $A$ and $B$ are independent, $P(X\mid A)=P(Y\mid A)$ and $P(X\mid B)=P(Y\mid B)$ since all four are $\frac12$, but $P(X\mid A,B)=1$ while $P(Y\mid A,B)=0$.
Thus, $P(X\mid A)$ and $P(X\mid B)$ do not determine $P(X\mid A,B)$, even assuming that the events $A$ and $B$ are independent.