Clutter Untyings, Correlation Inequalities, and Bounds for Combinatorial Reliability

We consider the clutter approach to the theory of reliability of stochastic binary systems. We propose the clutter untying transformations that are special inversions to one of the factor clutter transformations introduced by McDiarmid. We show that the untying transformations underlie the correlation inequalities conceptually related with the ideas of Esary and Proshan that are basic for the theory of reliability of stochastic binary systems. We give new proofs for these correlation inequalities (and characterize the statistical independence of monotone events in terms of the supports of their clutters), using only the untying transformation and not invoking (which is the practice now) the FKG-inequality for distributive lattices. As to stochastic binary system reliability itself, we show that the untying transformations generate a class of new bounds for this reliability, which are tighter than the classical Esary–Proshan bounds. One of these bounds is formulated in terms of a clutter and an antiblocking set for this clutter.