Upper bounds for the probability of a union by multitrees

The problem of finding bounds for P(A(1) boolean OR...boolean OR A(n)) based on P(A(k1) boolean AND... boolean AND A(ki)) (1 less than or equal to k(1) < ... < k(i) less than or equal to n, i = 1, ..., d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prekopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented.