Lower Bounds on the Probability of a Finite Union of Events

TL;DR

This paper develops new and optimal lower bounds for the probability of a union of events using linear programming, improving upon existing bounds and providing practical numerical examples.

Contribution

It introduces a numerically optimal lower bound via LP and a new analytical bound that outperforms previous results, advancing probability union bounds.

Findings

Optimal lower bound obtained through LP with N^2-N+1 variables.

New analytical lower bound surpasses Kuai et al.'s bound.

Numerical examples demonstrate the bounds' effectiveness.

Abstract

In this paper, lower bounds on the probability of a finite union of events are considered, i.e. $P\left(\bigcup_{i=1}^N A_i\right)$, in terms of the individual event probabilities $\{P(A_i), i=1,\ldots,N\}$ and the sums of the pairwise event probabilities, i.e., $\{\sum_{j:j\neq i} P(A_i\cap A_j), i=1,\ldots,N\}$. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the $P(A_i)$'s and $\sum_{j:j\neq i} P(A_i\cap A_j)$'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with $N^2-N+1$ variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, et al.; (iii) numerical examples are provided to illustrate the performance of the bounds.