A reference set, or a fiber, of a contingency table is the space of allrealizations of the table under a given set of constraints such as marginaltotals. Understanding the geometry of this space is a key problem in algebraicstatistics, important for conducting exact conditional inference, calculatingcell bounds, imputing missing cell values, and assessing the risk of disclosureof sensitive information. Motivated primarily by disclosure limitation problems where constraints cancome from summary statistics other than the margins, in this paper we study thespace $\mathcal{F_T}$ of all possible multi-way contingency tables for a givensample size and set of observed conditional frequencies. We show that thisspace can be decomposed according to different possible marginals, which, inturn, are encoded by the solution set of a linear Diophantine equation. Wecharacterize the difference between two fibers: $\mathcal{F_T}$ and the spaceof tables for a given set of corresponding marginal totals. In particular, wesolve a generalization of an open problem posed by Dobra et al. (2008). Ourdecomposition of $\mathcal{F_T}$ has two important consequences: (1) we derivenew cell bounds, some including connections to Directed Acyclic Graphs, and (2)we describe a structure for the Markov bases for the space $\mathcal{F_T}$ thatleads to a simplified calculation of Markov bases in this particular setting.
展开▼
