All examples I've seen for predicting $P(B|A_1 & A_2 ldots & A_n)$ assume that $A_1$, $A_2$, etc., are all mutually exclusive. What if I want to predict $B$ but have overlapping $A$'s?
For example:
B = it rains given A1 = there are clouds in the sky A2 = humidity level is over 80% A3 = it is currently "the rainy season" A4 etc Obviously, all of these $A$'s are not mutually exclusive. How does one calculate $P(B|A_1 & A_2 ldots & A_n)$ in this case?
Best Answer
To specifically answer your last question:
$$P(B|A_1,A_2, …, A_n) = frac{P(B, A_1, A_2, …,A_n)}{P(A_1, A_2, …,A_n)}$$
If you don't assume independence, then you have to calculate $P(B, A_1, A_2, …,A_n)$ and $P(A_1, A_2, …,A_n)$ both of which are joint distributions.
If you assume independence among the $A_i$'s then:
$$P(A_1, A_2, …,A_n) = prod_{i=1}^{n} P(A_i)$$
If you assume $B$ is conditionally independent given $A_i$'s then you can write:
$$begin{aligned}P(B, A_1, A_2, …,A_n) & = P(A_1, …, A_n | B)P(B) \ & = P(A_1|B)…P(A_n|B)P(B)end{aligned}$$
Both of the above calculations are usually tractable and/or computationally much less expensive.