Tight Probability Bounds with Pairwise Independence

TL;DR

This paper derives tight probability bounds for sums of pairwise independent Bernoulli variables, providing closed-form solutions and analyzing their applications in correlation and optimization problems.

Contribution

It introduces the first tight upper bound for the union probability with pairwise independence and extends the analysis to lower bounds and bounds for sums with larger thresholds.

Findings

Closed-form tight upper bound for union probability when k=1.

Ratio of Boole bound to tight bound is at most 4/3, and this is tight.

New bounds improve upon existing bounds in specific probability configurations.

Abstract

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\mathbf{p} \in [0,1]^n$. To prove the result, we show the existence of a positively correlated Bernoulli random vector with transformed bivariate probabilities, which is of independent interest. Building on this, we show that the ratio of the Boole union bound and the tight pairwise independent bound is upper bounded by $4/3$ and that the ratio is attained. Applications of the result in correlation gap analysis and distributionally robust bottleneck optimization are discussed. The result is extended to find the tightest lower bound on the probability of the intersection of $n$ pairwise independent events. Secondly, for any $k \geq 2$ and input marginal probability vector $\mathbf{p} \in [0,1]^n$, new upper bounds are derived by exploiting ordering of probabilities. Numerical examples are provided to illustrate when the bounds provide improvement over existing bounds. Lastly, we identify specific instances when the existing and the new bounds are tight, for example, with identical marginal probabilities.