Model-Independent Bounding of the Supports of Boolean Formulae in Binary Data

Data mining algorithms such as the Apriori method for finding frequent sets in sparse binary data can be used for efficient computation of a large number of summaries from huge data sets. The collection of frequent sets gives a collection of marginal frequencies about the underlying data set. Sometimes, we would like to use a collection of such marginal frequencies instead of the entire data set (e.g. when the original data is inaccessible for confidentiality reasons) to compute other interesting summaries. Using combinatorial arguments, we may obtain tight upper and lower bounds on the values of inferred summaries. In this paper, we consider a class of summaries wider than frequent sets, namely that of frequencies of arbitrary Boolean formulae. Given frequencies of a number of any different Boolean formulae, we consider the problem of finding tight bounds on the frequency of another arbitrary formula. We give a general formulation of the problem of bounding formula frequencies given some background information, and show how the bounds can be obtained by solving a linear programming problem. We illustrate the accuracy of the bounds by giving empirical results on real data sets.