I know that the chain rule says:
$P(A_1, A_2, A_3) = P(A_1 | A_2, A_3) P(A_2 | A_3) P(A_3)$
Is it then correct to say that
$P(A_1, A_2, A_3 | A_4) = P(A_1 | A_2, A_3, A_4) P(A_2 | A_3, A_4) P(A_3 | A_4)$
Or should I multiply by $P(A_4)$ at the end?
2026-03-29 04:49:02.1774759742
Chain rule for already conditioned probability
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You can find the correct form using Bayes' theorem: $P(A_1,A_2,A_3|A_4) = \dfrac{P(A_1,A_2,A_3,A_4)}{P(A_4)}$. So you would need $P(A_1,A_2,A_3,A_4)$ which is equal to $P(A_1 | A_2, A_3,A_4) P(A_2 | A_3,A_4) P(A_3|A_4) P(A_4)$.
This means that $P(A_1,A_2,A_3|A_4) = P(A_1 | A_2, A_3,A_4) P(A_2 | A_3,A_4) P(A_3|A_4)$.