Probability bounds for $n$ random events under $(n-1)$-wise independence

TL;DR

This paper characterizes probability measures for collections of $n$ events that are $(n-1)$-wise independent and provides efficiently computable bounds on the probability that at least $k$ of these events occur.

Contribution

It introduces a complete characterization of measures for $(n-1)$-wise independent events and derives sharp, polynomial-time computable bounds on occurrence probabilities.

Findings

Characterization of all measures with $(n-1)$-wise independence

Sharp bounds on probability of at least $k$ events occurring

Bounds are computable in polynomial time

Abstract

A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.