  This paper develops upper and lower bounds for the probability of Boolean
expressions by treating multiple occurrences of variables as independent and
assigning them new individual probabilities. Our technique generalizes and
extends the underlying idea of a number of recent approaches which are
varyingly called node splitting, variable renaming, variable splitting, or
dissociation for probabilistic databases. We prove that the probabilities we
assign to new variables are the best possible in some sense.
