Computing the probability of intersection

TL;DR

This paper presents a polynomial-time method to approximate the probability of the intersection of many dependent events in product probability spaces, based on limited information about their joint probabilities.

Contribution

It introduces conditions under which the intersection probability can be efficiently approximated using only small subset joint probabilities.

Findings

Approximation within relative error $oxed{ ext{epsilon}}$ achievable in polynomial time.

Requires each event probability to be below a specific threshold related to dependency degree.

Uses only small subset joint probabilities to compute the intersection probability efficiently.

Abstract

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 $k = e^{O(\Delta)} \ln (n/\epsilon)$.