I have a question which comes from an example of a textbook, but all I am concerned with is how we go from having probabilities $P(X|A)$ and $P(X|B)$ to having $P(X|A,B)$. In the example events $A$ and $B$ are independent, each influencing $X$. The example comes from Pattern Classification by Duda et. al, edition 2.
The example gives us a table defining $P(x_i|a_j)$ and $P(x_i|b_j)$. Shortly thereafter the authors have a summation over $P(x_1|a_i,b_j)$:
$$ \sum\limits_{i,j}P(x_1|a_i, b_j) $$
Later, they expand this sum into the particular values, which appear to me to be the values of $P(x_i|a_j)$. The numbers end up adding up to 4 which makes sense since there are four rows in $P(x_i|a_j)$ (each which must sum to 1).
The probability $P(x_1|a_i, b_j)$ makes a second appearance in the example, but unfortunately the authors have left a magic step and I am unsure of the values they used.
My question is why is this OK to do? What is the relationship between $P(X|A)$, $P(X|B)$, and $P(X|A,B)$? Do we need more information to get $P(X|A,B)$ and are they doing some sort of approximation?
The section is dealing with Bayesian belief networks. If you happen to have the book, it is Example 4 in Section 2.11.