Optimal Upper and Lower Bounds for Boolean Expressions by Dissociation
  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. 
Wolfgang Gatterbauer, Dan Suciu
cs.AI cs.DB cs.LO