Bonferroni inequalities

Bonferroni inequalities bound the probability of a union of events by truncating the inclusion–exclusion formula at successive terms, with alternating truncations giving upper and lower bounds.

Union bound: The inequality that the probability of a union is at most the sum of the individual event probabilities. The first Bonferroni upper bound is the union bound.
Probability space: A mathematical model consisting of outcomes, measurable events, and a probability measure. The inequalities apply to events measured within a probability space.
Reliability theory: The study of whether systems perform their required functions over time. Failure of any component forms a union whose probability can be bounded from component intersections.
Carlo Emilio Bonferroni: An Italian mathematician known for work in probability, statistics, and inequalities. His name became attached to the alternating bounds for probabilities of unions.
Pairwise intersection: The intersection of two sets or events, recording outcomes shared by that pair. Subtracting pairwise intersections tightens the first upper bound.
Event (probability theory): A measurable set of outcomes to which a probability can be assigned. Each set in the union is an event whose probability contributes to the bounds.
Poisson approximation: Methods that approximate distributions by a Poisson law, especially for rare events. Bonferroni bounds control errors when estimating the chance that at least one rare event occurs.
Inclusion–exclusion principle: A formula expressing the probability or size of a union through sums of intersections, with alternating signs. The inequalities are truncation bounds derived directly from this formula.
Triple intersection: The intersection shared by three sets or events. Adding triple intersections produces the next lower bound in the alternating sequence.
Finite additivity: The property that the probability of a finite disjoint union equals the sum of its probabilities. Inclusion–exclusion derives union probabilities by accounting for overlap rather than assuming additivity.

Linked from 2 pages

Boole's inequalityBroader topic: They sharpen Boole's bound by accounting for selected event intersections.