Computing the probability of intersection

Let $\Omega_1, \ldots, \Omega_m$ be probability spaces, let $\Omega=\Omega_1 \times \cdots \times \Omega_m$ be their product and let $A_1, \ldots, A_n \subset \Omega$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in \Omega$, $x=\left(\xi_1, \ldots, \xi_m\right)$, and that for each event $A_i$ there are $\Delta_i$ of other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\Delta=\max\{5,\ \Delta_i: i=1, \ldots, n\}$ and let $\mu_i=\min\{r_i,\ \Delta_i+1\}$ for $i=1, \ldots, n$. We prove that if $P(A_i) < (3\Delta)^{-3\mu_i}$ for all $i$, then for any $0 < \epsilon < 1$, 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 $\epsilon$ 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 $k = e^{O(\Delta)} \ln (n/\epsilon)$.