Computing the probability of intersection

Abstract

Let $Ω_1, \ldots, Ω_m$ be probability spaces, let $Ω=Ω_1 \times \cdots \times Ω_m$ be their product and let $A_1, \ldots, A_n \subset Ω$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in Ω$, $x=\left(ξ_1, \ldots, ξ_m\right)$, and that for each event $A_i$ there are $Δ_i$ of other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $Δ=\max\{5,\ Δ_i: i=1, \ldots, n\}$ and let $μ_i=\min\{r_i,\ Δ_i+1\}$ for $i=1, \ldots, n$. We prove that if $P(A_i) < (3Δ)^{-3μ_i}$ for all $i$, then for any $0 < ε< 1$, the probability $P\left( \bigcap_{i=1}^n \overline{A}_i\right)$ of the intersection of the complements of all $A_i$ can be computed within relative error $ε$ in polynomial time from the probabilities $P\left(A_{i_1} \cap \ldots \cap A_{i_k}\right)$ of $k$-wise intersections of the events $A_i$ for $k = e^{O(Δ)} \ln (n/ε)$.