Lower Bounds for the Probability of a Union via Chordal Graphs
Klaus Dohmen
TL;DR
The paper develops new Bonferroni-type lower bounds for the probability of a union of finitely many events by restricting the intersection terms to the clique complex of a chordal graph G. It proves a lower bound of the form P(union) ≥ (1/α(G)) times an alternating sum over cliques in C(G) up to size 2r, and shows this bound recovers classical results when G is complete while offering tighter, structure-aware estimates in general. The authors establish the key topological underpinnings via the contractibility of clique complexes of connected chordal graphs and bounds on Euler characteristics, and they demonstrate the necessity of chordality by presenting counterexamples for non-chordal graphs. Additionally, they present specialized lower bounds for trees (Hunter-type) and two-index generalizations, expanding the toolkit for probability inequalities under structured dependency graphs.
Abstract
We establish new Bonferroni-type lower bounds for the probability of a union of finitely many events where the selection of intersections in the estimates is determined by the clique complex of a chordal graph.