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?

Contents

$$P(B|A_1,A_2, …, A_n) = frac{P(B, A_1, A_2, …,A_n)}{P(A_1, A_2, …,A_n)}$$

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.

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