On bounding the union probability

We present new results on bounding the probability of a finite union of events, equation for a fixed positive integer N, using partial information on the events joint probabilities. We first consider bounds that are established in terms of {P(A_i)} and {Σ_jc_jP(A_i ∩ A_j)} where c₁, ..., c_N are given weights. We derive a new class of lower bounds of at most pseudo-polynomial computational complexity. This class of lower bounds generalizes the recent bounds in [1], [2] and can be tighter in some cases than the Gallot-Kounias [3]-[5] and Prékopa-Gao [6] bounds which require more information on the events probabilities. We next consider bounds that fully exploit knowledge of {P(A_i)} and {P(A_i ∩ A_j)}. We establish new numerical lower/upper bounds on the union probability by solving a linear programming problem with equation variables. These bounds coincide with the optimal lower/upper bounds when N ≤ 7 and are guaranteed to be sharper than the optimal lower/upper bounds of [1], [2] that use {P(A_i)} and {Σ_j P(A_i ∩ A_j)}.