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Boole-Bonferroni Inequalities and Linear Programming

We present a method to obtain sharp lower and upper bounds for the probability that at least one out of a number of events in an arbitrary probability space will occur. The input data are some of the binomial moments of the occurrences, such as the sum of the probabilities of the individual events, or the sum of the joint probabilities of all pairs of events. We develop a special, very simple linear programming algorithm to obtain these bounds. The method allows us to compute good bounds in an optimal way, utilizing only the first few terms in the inclusion-exclusion formula. Possible applications include obtaining bounds for the reliability of a stochastic system, solving algorithmically some stochastic programming problems, and approximating multivariate probabilities in statistics. In a numerical example we approximate the probability that a Gaussian process runs below a given level in a number of consecutive epochs.