Estimate $P(A_1|A_2 \cup A_3 \cup A_4...)$, given $P(A_i|A_j)$

Question

This question is related to some undergraduate research on summary generation of documents of which I am a part of. I am trying to estimate $P(A_1|A_2 \cup A_3 \cup A_4...A_k)$, where I know the values $P(A_i|A_j)\ \forall i,j \in\{1,2,...,n\}$. I understand that it is not possible to evaluate this probability exactly. Are there methods that relate to the approximation of such an expression under certain assumptions? Eg: Assuming event $A_2,A_3$ are independent. I would be glad if someone could point me to such resources. (webpages,books,papers,etc)

Answer 1

Hm... maybe we reduce the problem to: $$ Pr(A_{1}|A_{2}\cup A_{3}\cup...)=\frac{Pr(A_{1}\cap (A_{2}\cup A_{3}\cup...))}{Pr(A_{2}\cup A_{3}\cup...)} $$ Which is then... $$ Pr(A_{1}|A_{2}\cup A_{3}\cup...)=\frac{Pr((A_{1}\cap A_{2})\cup (A_{1}\cap A_{3})\cup (A_{1}\cap A_{4})...))}{Pr(A_{2}\cup A_{3}\cup...)} $$ And if... $$ Pr(A_{2}\cup A_{3}\cup...)\approx 1 $$ then... $$ Pr(A_{1}|A_{2}\cup A_{3}\cup...)\approx\Pr((A_{1}\cap A_{2})\cup (A_{1}\cap A_{3})\cup (A_{1}\cap A_{4})...)) $$ Which really depends on the mutual information (i.e. overlap in the venn diagram) between $A_{1}$ and the other events.

Answer 2

Maybe the principle of inclusion-exclusion

https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle?wprov=sfla1

or the Bonferroni inequalites

https://en.wikipedia.org/wiki/Boole%27s_inequality?wprov=sfla1

can help you to establish a connection between the expression and the set of given conditional probabilities.