On Bounding the Union Probability Using Partial Weighted Information

TL;DR

This paper introduces new lower bounds for the probability of a union of events using partial weighted information, improving upon existing bounds with computationally efficient methods.

Contribution

It presents two novel classes of lower bounds for union probabilities that generalize previous bounds and are computationally efficient.

Findings

Bounds are tighter than some existing bounds in certain cases.

New bounds are computationally feasible for fixed N.

The methods generalize and improve upon prior bounds.

Abstract

Effective bounds on the union probability are well known to be beneficial in the analysis of stochastic problems in many areas, including probability theory, information theory, statistical communications, computing and operations research. In this work we present new results on bounding the probability of a finite union of events, ${P\left(\bigcup_{i=1}^N A_i\right)}$, for a fixed positive integer ${N}$, using partial information on the events in terms of ${\{P(A_i)\}}$ and ${\{\sum_j c_j P(A_i\cap A_j)\}}$ where ${c_1}$, ${\dots}$, ${c_N}$ are given weights. We derive two new classes of lower bounds of at most pseudo-polynomial computational complexity. These classes of lower bounds generalize the existing bound in \cite{Kuai2000} and recent bounds in \cite{Yang2014,Yang2014ISIT} and are numerically shown to be tighter in some cases than the Gallot-Kounias bound \cite{Gallot1966,Kounias1968} and the Pr{\'e}kopa-Gao bound \cite{Prekopa2005} which require more information on the events probabilities.