Solved – Conditional probability given multiple non-mutually exclusive prior events

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?

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.

Similar Posts:

Rate this post

Leave a Comment