Conditional joint probabiltiy P(A|BCDEF) from single conditional P(A|B), P(A|C), etc?

Question

I have a matrix of conditional single probabilities, e.g.: P(A|D), P(A|E)...P(C|H); also, P(A) + P(B) + P(C) = 1.

How would I express a joint conditional probability, say P(A|DEFGH), if DEFGH are all independent events?

Via research I've seen many very helpful examples of Bayes' rule using 2 variables (A,D), e.g. P(A|D) = P(AD)/P(D), but am not sure how to generalize to multiple conditions?

P(A) = 0.0733, P(B) = 0.2099, P(C) = 0.7169 

      A      B       C
D [0.7436 0.1302 0.0081]
E [0.8757 0.4916 0.0759]
F [0.8764 0.2907 0.0041]
G [0.5940 0.3245 0.0134]
H [0.9959 0.7803 0.2625] 

Answer 1

Your example $P(A|DEFGH)$. Using the multiplication rule \begin{align*} P(ADEFGH) &= P(H)P(ADEFG|H) \\ &= P(H)P(G|H)P(ADEF|GH)\\ &=\dotsb\\ &=P(H)P(G|H)P(F|GH)P(E|FGH)P(D|EFGH)P(A|DEFGH)\\ &=P(H)P(G)P(F)P(E)P(D)P(A|DEFGH) \end{align*} where the last line is true because of independence. Thus $$P(A|DEFGH) = \frac{P(ADEFGH)}{P(H)P(G)P(F)P(E)P(D)}.$$