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

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)

• Do you also know the probabilities of $P(A_i)$ individually? – owen88 Jan 20 '18 at 9:07
• No, I do not. If I did, then you get a formula for $P(A|B_1\cup B_2...\cup B_k)$ and the problem is trivial. – bat_of_doom Jan 20 '18 at 13:15
• For a fixed $j$, how many $P(A_j|A_i)$ are non zero ? Your approach might still work if the underlying linear system is very sparse. – yultan Jan 20 '18 at 15:28
• Ah sorry, that approach was nonsense. $A_i$s are not disjoint! – bat_of_doom Jan 20 '18 at 15:42

## 2 Answers

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.

• The assumption $Pr(A_2\cup A_3...)=1$ won't work in my case. Plus, the final expression you gave is not really much of a reduction to an expression using just $Pr(A_i|A_j)$. I understand that evaluating this expression exactly without more information is not possible. But I need a reasonable estimate. – bat_of_doom Jan 20 '18 at 16:10
• Right, yeah I thought it wasn't much, but I think you need more information for any more significant improvements in simplification. I mean if we had independence between events or mutual exclusivity this would help. But assuming either doesn't make sense given the problem formulation. – Campbell Jan 20 '18 at 17:45

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.

• These inequalities are interesting, but the answer is not complete. Please also add how do I use the Bonferroni inequalities for k>2, as I only know P(A|B). For k=2, I suppose I could implement an approximation for $P(A_i)$ within the program, and then the Bonferroni bound for k=2 is straightforward. – bat_of_doom Jan 24 '18 at 8:24