Computing the probability of intersection

Let ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_m$ be probability spaces, let $\mathrm{\Omega }={\mathrm{\Omega }}_1\times \cdots \times {\mathrm{\Omega }}_m$ be their product and let $A_1,\dots ,A_n\subset \mathrm{\Omega }$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x\in \mathrm{\Omega }$, $x=({\xi }_1,\dots ,{\xi }_m)$, and that for each event $A_i$ there are ${\mathrm{\Delta }}_i$ of other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\mathrm{\Delta }=\max \{5,{\mathrm{\Delta }}_i:i=1,\dots ,n\}$ and let ${\mu }_i=\min \{r_i,{\mathrm{\Delta }}_i+1\}$ for $i=1,\dots ,n$. We prove that if $P(A_i)<(3\mathrm{\Delta }{)}^{-3{\mu }_i}$ for all $i$, then for any $0<\epsilon <1$, the probability $P\left({\bigcap }_{i=1}^n{\bar{A}}_i\right)$ of the intersection of the complements of all $A_i$ can be computed within relative error $\epsilon$ in polynomial time from the probabilities $P(A_{i_1}\cap \dots \cap A_{i_k})$ of $k$-wise intersections of the events $A_i$ for $k=e^{O(\mathrm{\Delta })}\ln (n\mathrm{/}\epsilon )$.