Cell Bounds in k-way Tables Given Conditional Frequencies

For data summarized and released as a contingency table, considerable attention has been accorded to cell bounds given marginal totals. Here, we consider bounds on cell counts for k-way tables when observed conditional probabilities and total sample size are released. If this information implies narrow bounds, a disclosure risk may result. We compute sharp integer bounds using integer programming and demonstrate that, in some cases, they can be unacceptably narrow. We also derive closed-form solutions for linear relaxation bounds, and show that they can be improved via a method that can also account for rounding uncertainty. The gaps between the sharp bounds and those of their linear relaxations are often large, which implies the utility of the latter is limited, especially if the sharp bounds can be computed quickly. Our formulations can solve small tables with small sample sizes quite quickly, but large instances can take on the order of hours. Adapted from the source document.